extension: trigonometric functions and their...

Chapter 8

Extension: TrigonometricFunctions and Their Derivatives

MODELLING

•

M AT H

•

Many phenomena in nature are periodic, and so can be modelled using combinationsof sine and cosine functions, which are the basic periodic functions. The rhythms ofEarth, such as its daily rotation, the seasons, the tides, weather, and so on, can all bemodelled using trigonometric functions. Many of our bodily rhythms can also bemodelled using trigonometric functions. Examples are the rhythms of our heart beats,our breathing, the concentration of various substances in our bloodstreams, and theelectrical and chemical rhythms in our brains. But trigonometric functions are used formany things other than modelling periodic phenomena. Wherever angles are essentialin problems, such as when a baseball is hit and its initial path is inclined at a particularangle, trigonometric functions are helpful.

1. Distance between two points Determine thedistance between the points in each pair.a) (1, 2) and (3, 5)b) (�2, 4) and (1, �7)

2. Even functions (Section 1.1) a) State thedefinition of an even function.b) Determine whether each function is even.i) f (x) � x2

ii) f (x) � x2 � x3

iii) f (x) � x2 � x4

iv) f (x) � cos x

3. Odd functions (Section 1.1) a) State thedefinition of an odd function.b) Determine whether each function is odd.i) f (x) � x3

ii) f (x) � x3 � x2

iii) f (x) � x � x3

iv) f (x) � sin x

4. Circle geometry A circular arc has length3 cm, and the radius of the circle is 2 cm.Determine the measure of the angle subtendedby the arca) in radiansb) in degrees

5. Circle geometry Assume that Earth movesaround the sun at a constant speed in a circle ofradius 150 000 000 km.a) What angle does Earth’s path subtend in oneday?b) How far does Earth travel in one day?c) What is the Earth’s speed of motion aroundthe sun in kilometres per hour?

6. Circle geometry As Earth rotates on its axis,each point on Earth travels in a circle. Theradius of the circle depends on the latitude ofthe point.a) Look up the latitude of your location onEarth.b) Assuming that Earth is a sphere of radius 6400 km, calculate the radius of yourpath.c) Determine the circumference of your path.d) Determine the speed at which your location moves around Earth (in kilometres per hour).

7. Angle measures Write each angle in degrees.a) � radb) rad

c) rad

d) rad

e) rad

f ) 1 rad

8. Angle measures Write each angle in radians.a) 15�b) 75�c) 105�d) 120�e) 150�f ) 700�

9. Trigonometric functions Determine thevalues of sin �, cos �, and tan �.

10. Trigonometric functions Use appropriateright-angled triangles to determine exact valuesfor the following quantities.

b)

–1–2

1

2

–1

0

y

x

(–2, 1)

�

a)

2 4

2

4

–2

0

y

x�

(3, 4)

–2

�6

�4

�3

�2

488 MHR Chapter 8

Review of Prerequisite SkillsReview of Prerequisite Skills

Web ConnectionTo review trigonometry interactively, go towww.mcgrawhill.ca/links/CAF12 and follow the links.

� sin � cos � tan �

30��or �6

45��or �4

60��or �3

11. Trigonometric functions Use a calculator todetermine the sine, cosine, and tangent of thefollowing angles, rounded to four decimalplaces.a) 27.7�b) 81.4�c) 0.8 radd) rad

12. Trigonometric functions Use your calculatorto determine an angle, in radians, rounded totwo decimal places, for whicha) sin � � 0.34b) cos � � 0.6c) tan � � 4

13. Trigonometric functions If sin � � 0.5

and , determine tan � without

determining �.

14. Trigonometric identities Draw a unit circlecentred at the origin O. Label a point A on thecircle in the first quadrant. Drop a perpendicularfrom A to intersect the x-axis at B. Label �AOBas �.a) Consider the right-angled triangle OAB, and use the Pythagorean theorem to prove thatsin2

� � cos2� � 1.

b) Is there an angle � for which sin � � 0.8 andcos � � 0.2? If so, determine the angle. If not,explain why not.

15. Determining angles Draw a unit circlecentred at the origin.a) Sketch an angle � in the first quadrant,� � [0, ]. Then, sketch the related angles

��, � � �, 2� � �, and � � �.b) Repeat part a) if the starting angle � is in thesecond quadrant, � � [ , �].c) Repeat part a) if the starting angle � is in the

third quadrant, � � [�, ].d) Repeat part a) if the starting angle � is in the

fourth quadrant, � � [ , 2�].16. Determining coordinates of points For eachdiagram, determine the coordinates of the point A.

17. Determining angles For each diagram,determine the angle �, in radians.

18. Graphing trigonometric functions Sketch agraph of two cycles of each function.a) y � sin xb) y � cos xc) y � tan xd) y � sin (x � )e) y � 3 cos (2x)

19. Solving trigonometric equations Solve eachequation for � � [0, 2�]. Round your answers tothe nearest hundredth, if necessary.a) 3 cos � � cos � � 2b) 2 sin2

� � sin � � 1 � 0c) 6 sin2

� � sin � � 5 � 0d) sin2

� � sin � � 6 � 0

�3

c) d)

–2–4

–2

–4

0

y

�

(–4, –2)

x 2 4

–2

–4

0

y

x�

(5, –3)

b)a)

2

4

0

y

x�

(3, 2)

2 4 –2–4

2

4

0

y

x�

(–3, 4)

c) d)y

x

3

0

911�

A

y

x

6

0

A

5�3

b)a)

x50°

5

A

0

y y

x

125°4

A

0

32�

32�

�2

�2

cos � = 32

�7

Review of Prerequisite Skills MHR 489

20. Slopes of lines; Trigonometric functionsa) Sketch the line joining the origin with thepoint (3, 4).b) Determine the slope of the line in part a).c) Determine the tangent of the angle that theline in part a) makes with the x-axis.d) Compare the results of parts b) and c). Whatconclusion can you draw?e) Is the conclusion you made in part d) truefor all lines through the origin? If so, provide aconvincing argument. If not, find an example forwhich it is not true.

The Reciprocal Trigonometric FunctionsThe following reciprocal trigonometricfunctions are defined.

Cosecant:

Secant:

Cotangent:

21. Trigonometric identities Rewrite eachexpression so that no denominators remain.

a)

b)

c)

d)

22. Trigonometric identities Simplify eachexpression so that no denominators remain.a)

b)

c)

23. Trigonometric identities Write eachexpression in terms of sine and cosine only.a) sec x csc xb) tan x sec xc) sec2xd) cot x sec x

24. Trigonometric identities Show that eachidentity is valid.

a)

b)

c)

25. Trigonometric functions Determine the exactvalue of each quantity.

a) csc

b) sec

c) cot

d) sec 0e) csc 0f ) cot �g) csc 30�h) sec 90�i) cot 60�

26. Trigonometric identities Consider theidentity sin2

� � cos2� � 1.

a) Divide each term of the identity by sin2� to

obtain another relation. Express this relationusing the reciprocal trigonometric functions.For which values of � is the resulting relationvalid?b) Divide each term of the identity by cos2

� toobtain another relation. Express this relationusing the reciprocal trigonometric functions.For which values of � is the resulting relationvalid?

�4

�4

�3

sectan

csc�

��=

cotcscsec

��

�=

cotcossin

��

�=

sinsin

xx3

sintan

xx

tansin

xx

12

43cos sinx x

−

1 3tan cosx x

−

1 12 2sin cosx x

+

1 1sin cosx x

+

cottan

��

= 1

seccos

��

= 1

cscsin

��

= 1

490 MHR Chapter 8

8.1 Addition and Subtraction Formulas MHR 491

Addition and Subtraction Formulas

The main goal of this chapter is to determine the derivatives of the basic trigonometricfunctions. In order to use first principles to find the derivative of the function f (x) � sin x,we need to determine the following limit.

It may not be clear how to determine this limit, because we are unfamiliar with expandingexpressions such as sin(x � h). This requires an addition formula for the sine function.

In this section, we develop addition and subtraction formulas for the sine, cosine, andtangent functions. Starting with the sine function, we want a formula that expresses sin(x � h) in terms of sin x and sin h. In general, we want a formula for sin(a � b) in terms of sin a and sin b. Consider the diagram.

The diagram has been constructed so that �RQU, �RUT,�UPS, and �RST are all right angles. To simplify thecalculations, we scale the diagram so that the measure ofsegment RT is 1 unit. The general strategy for developing thedesired formulas is to express various segments in the diagramin terms of trigonometric functions of angles a, b, and a � b.Then, we use relations among the segments to determinerelations among the trigonometric functions.

To carry out this strategy, we need to determine �RUQ.

First note that, in �TUP, �TUP � � a. Then,

since �TUP � �RUQ � , it follows that �RUQ � a.

Note that in �TRS,

Also,RS � QP

� UP � QU

Thus, sin (a � b) � UP � QU

We can express UP and QU in terms of trigonometric functions of a and b.In �TRU,

cos TUTRTU

b =

= =(since TR 1)

sin RURTRU

b =

= =(since RT 1)

sin( ) RSRTRS

a b+ =

= =(since RT 1)

�2

�2

f xf x h f x

hx h x

h

h

h

�( ) lim( ) ( )

lim sin( ) sin( )

��

� � �

0

0

8.18.1

T S P

U

QR

ab a b+

1

492 MHR Chapter 8

In �TUP, In �RQU,

or or

UP � sin a cos b QU � cos a sin b

Since sin(a � b) � UP � QU, we have the addition formula for sine,

sin(a � b) � sin a cos b � cos a sin b

By following a similar strategy using the fact that TS � TP � SP, the following additionformula for cosine can be determined.

cos (a � b) � cos a cos b � sin a sin b

Once we have the basic identity sin(a � b) � sin a cos b � cos a sin b, we can obtain anidentity for the sine of the difference of two angles simply by substituting �b for b in theformula and recalling that sin(�b) � �sin b and cos(�b) � cos b. sin(a � b) � sin (a � [�b])

� sin a cos (�b) � cos a sin (�b)� sin a cos b � cos a (�sin b)� sin a cos b � cos a sin b

The formula for cos (a � b) can be obtained in the same way.cos(a � b) � cos(a � [�b])

� cos a cos (�b) � sin a sin (�b)� cos a cos b � sin a (– sin b)� cos a cos b � sin a sin b

The subtraction formulas for sine and cosine are

sin (a � b) � sin a cos b � cos a sin bcos (a � b) � cos a cos b � sin a sin b

Addition and subtraction formulas for tangent functions are obtained from the addition andsubtraction formulas for sine and cosine. For instance,

We would prefer to express tan (a � b) in terms of tan a and tan b, so we divide thenumerator and denominator by cos a cos b.

tan ( )

sin coscos cos

cos sincos cos

cos coscos cos

a b

a ba b

a ba b

a ba b

+ =+

−− sin sincos cos

a ba b

tan ( )sin ( )cos ( )

sin cos cos sincos cos sin si

a ba ba b

a b a ba b a

+ = ++

= +− nn b

cosQURUQUsin

a

b

=

=

sin UPTUUP

cos

a

b

=

=

Recalling that tan(�b) � �tan b, we can find the subtraction formula for tangent using thisaddition formula, as we did with sine and cosine.

Thus, the addition and subtraction formulas for tangent are

tan(a � b)

tan(a � b)

In Example 1, some of the addition formulas are verified with specific values. Testing values does not prove that the identities we derived are valid (for proof, we need the derivations above), but it is a useful habit to develop, as it is a good way to discover errors.

Example 1 Testing Addition and Subtraction Formulas

a) Test the formula cos(a � b) � cos a cos b � sin a sin b for a � , b � .

b) Test the formula sin(a � b) � sin a cos b � cos a sin b for a � , b � .

c) Test the formula tan(a � b) for a � 0.

Solutiona) Consider the left side and the right side separately and verify that they are equal.

L.S. R.S.= +

= +

=

= −

=cos( )

cos

cos

cos cosa b a b

� �

�

4 2

34

12

−−

=

−

=

sin sin

cos cos sin sin

a b

� � � �4 2 4 2

122

0 12

1

12

−

= −

=

( ) ( )

L.S. R.S.

= −+

tan tantan tan

a ba b1

�3

�6

�2

�4

� ��

�tan tantan tan

(tan tan )a ba b

a b1

1

� ��

tan tantan tan

(tan tana ba b

a b1

1)

tan ( ) tan [ ( )]

tan tan ( )tan tan ( )

tan tan

a b a b

a ba b

a b

− = + −

= + −− −

= −+

1

1 ttan tana b

=+

−

=

sincos

sincos

sincos

sincos

tan

aa

bb

aa

bb

1

aa ba b

+−

tantan tan1


b)

c) L.S. � tan (a � b)� tan (0 � b)� tan (�b)� � tan b

The addition and subtraction formulas can be useful tools for solving trigonometric equations.

Example 2 Using Addition and Subtraction Formulas to Solve TrigonometricEquations

Solve for x in the domain [0, 2�].

a) sin x cos � cos x sin � b) cos cos x � sin sin x � 1

c) � d) sin(x � ) � cos x

Solutiona) The left side of the equation, sin x cos � cos x sin , is in the pattern of the sin(a � b)

formula with a � x and b � , so it can be simplified to a single trigonometric function.

sin x cos � cos x sin �

sin(x � ) �We need to find x in the domain [0, 2�]. This means we need to find angles (x � ), in the

interval [� , ], whose sine is . The angles can be read from the graph of

the sine function. Remember to choose Radians from the Mode settings.

12

116��

6

�6

12

�6

12

�6

�6

�6

�6

�6

2�4

1 1+ tan tanxtan tanx − 1

�4

�4

12

�6

�6

L.S. R.S.=

L.S. R.S.=

494 MHR Chapter 8

R.S. = −+

= −+

= −+

= −

tan tantan tan

tan tantan tantan

ta

a ba b

bb

b

10

1 00

1 0nn b

R.S. = +

=

+

sin cos cos sin

sin cos cos s

a b a b� � �6 3 6

iin �3

12

12

32

32

14

34

1

=

+

= +

=

L.S. = +

=

=

sin

sin

� �

�

6 3

21

Window variables:

x� [� , ], y� [�1.5, 1.5]11

6��

6

x � � or x � �

x � x � �

The roots of the equation are and �.

b) The left side of the equation, cos cos x � sin sin x, is in the pattern of the

cos(a � b) formula with a � and b � x.

cos cos x � sin sin x � 1

cos( � x) � 1

We need to find x in the domain [0, 2�]. This means we need to find angles ( � x ), in the

interval [ , ], whose cosine is 1. The angles can be read from the graph of the cosine function.

There is only one value of � x in the interval [ , ] whose sine is 1.

� x � 2�

x �

The required root is .

c) tan x � tan 1 � 1 � tan x tan 1

Each side of the equation has a familiar pattern. The left side is the numerator of the tan (a � b) formula and the right side is the denominator. Divide both sides by( ).

tan (x � 1) � 1We need to find x in the interval [0, 2�]. This means we need to find angles (x � 1), in theinterval [�1, 2� � 1], that have a tangent of 1. The angles can be read from the graph of thetangent function.

tan tantan tan

(tan tan )xx

x−

+ = ≠ −11 1

1 1 1

1 1+ tan tanx

74�

74�

�4

94��

4�4

94��

4

�4

�4

�4

�4

�4

�4

�4

�3

�3

56��

6�6

�6


Window variables:

x� [0, ], y� [�1.5, 1.5]94�

Window variables:x� [�1, 2� � 1], y� [�1.5, 1.5]

There are two values of (x � 1) in the interval [�1, 2� � 1] that have a tangent of 1.

x � 1 � or x � 1 �

x � 1 � x � 1 �

� 1.7854 � 4.9270

The roots of the equation are approximately 1.7854 and 4.9270. If we test these roots, neither violates the restriction tan x tan 1 ��1.

d)

In the interval [0, 2�], x � or x � . The roots of the equation are and .54��

454��

4

54��

4

54��

4

496 MHR Chapter 8

Expand using the addition formula for sine.

Multiply both sides by .

Divide both sides by cos x.

2

sin cos cos sin cos

sin cos cos

si

x x x

x x x

� �4 4

2

12

12

2

��

��

��

��

� �

nn cos cos

sin cos

tan

x x x

x x

x

� �

�

�

2

1

Key Concepts� Addition and subtraction identities for sine, cosine, and tangent:

sin (a � b) � sin a cos b � cos a sin bsin (a � b) � sin a cos b � cos a sin bcos (a � b) � cos a cos b � sin a sin bcos (a � b) � cos a cos b � sin a sin b

Communicate Your Understanding1. Note that the sign on the right side of the identity for sin (a � b) must be �, since, if a � b, the left side of the equation is sin 0 � 0, so the right side of the equation mustbe identically zero (i.e., zero no matter what values a and b have). Explain why the signon the right side of the identity for sin (a � b) must be �.2. Using arguments like those of question 1, explain why the signs on the right sidesof the identities for the addition and subtraction formulas for cosine must be what they are.3. Repeat question 2 for tangent.

tan( ) tan tantan tan

a ba b

a b− = −

+1

tan( ) tan tantan tan

a ba b

a b+ = +

−1

sin cosx x��

��

�4

2

Practise1. Express as a single trigonometric function,then evaluate.a) sin 45� cos 15� � cos 45� sin 15�b) sin 45� cos 15� � cos 45� sin 15�c) cos 45� cos 15� � sin 45� sin 15�d) cos 45� cos 15� � sin 45� sin 15�

e)

2. Express as a single trigonometric function.a) sin A cos B � cos A sin Bb) sin M cos N � cos N sin Mc) cos A cos 2A � sin A sin 2Ad) sin A sin B � cos A cos Be) sin x sin y � cos x cos y

f )

g) cos2x � sin2xh) cos2x � sin2x

3. Expand. Then, determine the exact value ofeach quantity.

a) b)

c)

Apply, Solve, Communicate4. Application Solve for x where x� [0, 2�].a) sin x cos 2 � cos x sin 2 � �0.5b) cos x cos 1 � sin x sin 1 � 0.4c) cos 3x cos x � sin 3x sin x � 0d) sin 2x cos x � cos 2x sin x � 0e) 2 cos (2 � x) � � 0f ) sin 2x cos 3x � �1 � cos 2x sin 3xg) cos 2x cos x � sin 2x sin x

h) �

i) tan 3x � tan x � 3(1 � tan 3x tan x)j) cos 5x cos x � sin 5x sin x � 0.5

k) 4 sin 2x cos x � 4 cos 2x sin x � 2

l)

m) 6 sin 5x cos 3x � 3 � 6 cos 5x sin 3x

5. Inquiry/Problem Solving The two lines y ��2x � 4 and y � 3x � 6 have the same x-intercept. Let the angle between the first line and the x-axis be �� the angle between the second line and the x-axis be �� andthe acute angle between the two lines be �.a) How are the slopes of the lines related totan � and tan �? b) Express tan � in terms of the slopes of thelines. c) Determine the measures of the angles �, �,and �.

6. Application Develop the addition formula forcosine:cos (a � b) � cos a cos b � sin a sin bModel your development of this identity on the one used for the addition formula for sine.

7. Communication Show that each statement istrue.a) cos (� � x) � �cos xb) sin (� � x) � �sin xc) cos (� � x) � �cos xd) sin (� � x) � sin xe) tan (� � x) � tan xf ) tan (� � x) � �tan x

g) sin � �cos x

h) cos � �sin x− −

�2

x

32� +

x

2 4 6

2

4

–2

0

y6

x�

�

�

y x= 3 – 6

y x= 2 – 4

3 14 1 1

1(tan tan )( tan tan )

xx

+− =

3tan tan

tan tan

x

x

+

−

�

�3

13

3

tan � �4 6

+

cos � �2 3

−

sin � �3 4

+

21 2

tantan

xx−

tan tan

tan tan

� �

� �12 6

112 6

+

−


A

B

498 MHR Chapter 8

8. Simplify.

a)

b)

9. Suppose that x and y are in the interval

, sin x � , and cos y � .

a) Determine the value of cos x and sin y.b) Find the following.i) sin (x � y) ii) cos (x � y) iii) tan (x � y)

10. If x is in the interval , y is in the

interval , sin x � , and tan y � � ,

determine each value.a) sin (x � y) b) cos (x � y) c) tan (x � y)

11. Inquiry/Problem Solving a) Write the followingexpression as a sum of two terms, where eachterm is a product in which x does not appear inone factor and h does not appear in the otherfactor.

b) Repeat part a) for the expression

.

12. a) Derive the addition formula for secant.

b) Derive a similar formula for sec(a � b).

13. a) Derive the addition formula forcosecant.

b) Derive a similar formula for csc(a � b).

14. a) Derive the addition formula forcotangent.

b) Derive a similar formula for cot(a � b).

15. Communication Determine the conditions on angles x and y so that each equation issatisfied.a) sin (x � y) � sin (x � y)b) cos (x � y) � cos (x � y)c) sin x � sin y � cos x � cos y

16. a) Derive the following transformationformula for tangent.

b) Derive a transformation formula for tan x ��tan y similar to the one in part a).

tan tansin( )

cos cosx y

x yx y

+ = +

cot( )cot cotcot cot

a ba ba b

+ = −+

1

csc( ) csc csccot tan

a ba b

a b+ = +

sec( )sec sec

tan tana b

a ba b

+ = −1

sin( ) sin( )x h xh

+ −

cos( ) cos( )x h xh

+ −

34

35

��

2,�

��

02

, ��

��

513

450

2, ��

��

tan tan

tan

� �4 4

−

− +

x x

x

sin( ) cos( )sin

x xx

� � �30 60� �C

Trigonometry has its roots in early mathematical descriptions of the heavens. It was recognizedthat the apparent paths of stars and planets are often circles, or nearly circles. The earliest knowntable of values related to trigonometry was produced by the Greek mathematician Hipparchus(190–120 B.C.) in 150 BC. Ptolemy (85–165 A.D.) made use of identities that are equivalent toaddition formula for sine, the sine law, and several others. Arabic mathematicians worked withsines and cosines, and by 980 A.D. made use of the double angle formula for sine. The Arab wordjaib, meaning fold, was translated into Latin by European mathematicians as sinus, from which weget the modern word sine. In 1542 the Polish astronomer Copernicus (1473–1543) summarized allof the trigonometry relevant for astronomy.

Historical Bite: Early Trignometry

8.2 Double-Angle Formulas MHR 499

The double-angle formulas are formulas for sin 2a, cos 2a, and tan 2a. In this section, weuse the addition formulas for the sine, cosine, and tangent functions to obtain theseformulas. The double-angle formulas are quite useful for solving trigonometric equationsand arise frequently in applications of trigonometry.

Start with the addition formula for sine.sin (a � b) � sin a cos b � cos a sin b

Let b � a to obtainsin 2a � sin (a � a)

� sin a cos a � cos a sin awhich simplifies to the double-angle formula for sine:

sin 2a � 2 sin a cos a

The double-angle formula for cosine is obtained in the same way as that for sine. Set b � ain the addition formula for cosine:cos (a � b) � cos a cos b � sin a sin b

cos 2a � cos (a � a)� cos a cos a � sin a sin a� cos2a � sin2a

Using the Pythagorean identity sin2 a � cos2 a � 1, we can write the double-angle formula forcosine in two other forms. For instance, if we solve the Pythagorean identity for sin2 a, sin2 a � 1 � cos2 aand substitute the result into the double-angle formula, we getcos 2a � cos2 a � (1 � cos2a)

� cos2 a � 1 � cos2 a� 2cos2a � 1

Alternatively, we could solve the Pythagorean identity for cos2 a, cos2 a � 1 � sin2 aand substitute the result into the original version of the double-angle formula to getcos 2a � cos2 a � sin2 a

� 1 � sin2 a � sin2 a� 1 � 2sin2 a

Thus, we have three versions of the double-angle formula for cosine.

cos 2a � cos2 a � sin2 acos 2a � 2cos2 a � 1cos 2a � 1 � 2sin2 a

The three versions are equivalent, but, in particular situations, one form may be moreconvenient than the others.

Double-Angle Formulas 8.28.2

500 MHR Chapter 8

Example 1 Simplifying a Trigonometric Expression

Simplify . State any restrictions on the variable in the domain [0, 2�].

SolutionWe use the double-angle formulas for sine The denominator cannot be equal and cosine. to 0, so

Example 2 Solving Trigonometric Equations

Solve for x in the domain [0, 2�].a) 4 sin x cos x � 1 � 0 b) cos 2x � sin x � 0

Solutiona) 4 sin x cos x � 1 � 0Rewrite 4 sin x cos x using the double-angle formula sin 2a � 2 sin a cos a.2(2sin x cos x) � 1 � 0

2(sin 2x) � 1 � 0

Since x is restricted to the domain [0, 2�], 2x is restricted to the domain [0, 4�]. Thesolutions for 2x in this domain are

2x � or or or

Thus,

x � or or or

The roots of the equation are , , , and .

b) cos 2x � sin x � 0In trigonometric equations such as this one, where there are two different angles, it isusually a good strategy to make changes, if possible, so that only one angle appears. (Thisprocess is called matching arguments, since an angle is often called an argument of atrigonometric function.) Here we can use the double-angle formula to express cos 2x interms of trigonometric functions of x. The double-angle formula of cosine has threeversions from which to choose. We choose the one that matches the other term in theequation, sin x.

1712

�1312

�512��

12

1712

�1312

�512��

12

176�13

6�5

6��

6

sin 2 12

x =

1 2 0

2 1

22

32

52

72

434

54

74

�

cos

cos

, , ,

, , ,

x

x

x

x

� � � �

� � � �

sincos

sin cos( sin )

sin cossin

cossin

21 2

21 1 22

2

2

2

xx

x xx

x xx

x

− =− −

=

=xxx= cot

sincos

21 2

xx−


cos 2x � 1 � 2 sin2 xThus,(1 � 2 sin2 x) � sin x � 0

2 sin2 x � sin x � 1 � 0Factoring gives us

x � or x � or x �

The roots of the equation are , , and .

Having developed the double-angle formulas for sine and cosine, we now turn to thedouble-angle formula for tangent. It can be derived by starting with the addition formulafor tangent and setting b � a.

Example 3 Spotlight Position

A painting, 1 m tall, is to be hung 0.8 m down from the ceiling. A spotlight is to bemounted on the ceiling so that the angles marked � in the diagram are equal. Determinethe position on the ceiling where the spotlight must be mounted.

SolutionUsing the two right triangles in the diagram and the tangent ratio , we have

We can now substitute for tan �.

tan .

tan .

tantan

.

�

�

�

�

�

�

��

0 8

2 1 8

21

1 82

L

L

L (double-angle formula for tangent)

oppositeadjacent

tan tantan

(tan )2 21

12aaa

a=−

≠ ±

tan tan( )

tan tantan tantantan

2

12

1 2

a a a

a aa aaa

= +

= +−

=−

116�7

6��

2

116�7

6��

2

(sin )( sin )

sin sin

x x

x x

− + =

= = −

1 2 1 0

1 12

or

L

0.8 m

1 m

��

Spotlight

–1

1

0 x

y

2� �

23� 2�

y = 1

y = –0.5

y x= sin

502 MHR Chapter 8

The distance from the wall to the spotlight is 2.4 m. Note that we did not have to solve for � to find the solution to the problem. We needed only tan �.

2 0 8

1 0 8

1 8

1 6

0 641 8

1 60 64

2

2

2

2

.

.

.

.

..

..

L

L

L

LL

L

L

LL

−

=

−=

−==

===

1 8

0 2 1 152

5 76

2 4

2

2

.

. .

.

.

L

L

L

L

Key Concepts� Double-angle formula for sine

sin 2a � 2 sin a cos a� Double-angle formulas for cosine

cos 2a � cos2a � sin2 acos 2a � 2cos2 a � 1cos 2a � 1 � 2sin2 a

� Double-angle formula for tangent

Communicate Your Understanding1. Explain how to derive a double-angle formula starting from an addition formula.2. The strategy often used when solving trigonometric equations is to match arguments and express the equation in terms of one trigonometric function. Explainwhat this means and why the strategy is used.

tan tantan

(tan )2 21

12aaa

a=−

≠ ±


Practise1. Express as a single sine or cosine function.a) 10 sin x cos x b) 5 sin (2x) cos (2x)c) sin (6x) cos (6x) d)

e) f ) 2 cos2 5� � 1

g) h) 2 cos2 (3� � 2) � 1

2. Simplify each expression. State anyrestrictions on the variable in the domain [0, 2�].

a) b) 2 tan a cos2a

c) 2 sin2a � cos 2a

Apply, Solve, Communicate3. Inquiry/Problem Solving Expand using adouble-angle formula.a) 3 sin 4x b) 6 cos 6xc) 1 � cos 8x d) tan 4x

e)

4. Communication a) If you know the values

of sin and cos , how can you determine

the value of sin x without calculating x?

b) If you know the value of sin , is it

possible to determine the value of cos x withoutcalculating x? If so, explain how.

5. Solve for x in the domain [0, 2�].a) cos 2x � cos x � 1 � 0b) cos 2x � sin xc) 3 tan x � tan 2xd) sin x � 6 sin 2xe) sin 2x cos x � sin2 x � 1f ) sin 2x � sin x � 0g) sin 2x � cos 2x � 0h) 3 cos 2x � 2 � cos x � 0i) sin 2x � tan xj) 2 sin x cos x � cos 2xk) 3 sin 2x � cos x � 0l) 3 sin x � cos 2x � 2m) 5 � 13 sin x � 2 cos 2x

6. Two ropes (2 m and 3 m long) used tostabilize a pole for a volleyball net are anchored

to the ground. The angle between the two ropesis equal to the angle between the ground and thelower rope. Determine the distance from thebase of the pole to the point at which the ropesare anchored to the ground.

7. Application a) Express sec 2� in terms of sec �and tan �.b) Express csc 2� in terms of csc � and sec �.

8. Application Determine formulas for a) sin 3� in terms of sin �b) cos 3� in terms of cos �c) tan 3� in terms of tan �

9. Express sin 2� and cos 2� in terms of tan �.

10. Determine sin 2� if sin � � cos � � .

11. If cos 4� � cos 2� � , determine the

possible values of cos � in the domain [0, 2�].

12. Inquiry/Problem Solving A right circular cone isinscribed in a sphere of radius 30 cm. The semi-vertical angle of the cone is x, as shown.

a) Determine an expression for the volume ofthe cone in terms of x.b) When the volume of the cone is 9000� cm3,show that x satisfies the equation 8cos6x � 8cos4x � 1 � 0.c) Use graphing technology to solve theequation of part b). Explain what the solutionsmean.

x

centre

78

12

x

x

2 m

3 m

x2

x2

x2

cos sinsin

2 2x

xx

−

sincos

2aa

1 2 23

2− sin �

cos sin2 2

2 2� �−

42 2

sin cosx x

A

C

B

504 MHR Chapter 8

Limits of Trigonometric Functions

To determine the derivative of a trigonometric function using the definition of thederivative, we need to know certain basic limits of trigonometric functions. These limits are developed in this section.

The first limits we need are the limits of sin � and cos � as � approaches 0. Since sine andcosine are continuous functions, these limits are just the values of the functions at � � 0.

The next limit we need, one of the most important for trigonometric functions, is

Substituting � � 0 into the expression results in , which is undefined. In the following

investigation, you will use a calculator to get a sense of whether this limit exists, and if so,what its value might be. It is not clear whether to use degrees or radians, so you will use both.

Investigate & Inquire: An Important Trigonometric Limit1. Using angles in degrees, construct tables of values for for values of � approaching 0from the left and from the right.

2. Examine the tables in step 1. Does it appear that exists? If so, what

appears to be the value of the limit when � is in degrees?3. Repeat step 1 using angles in radians.

4. Examine the tables in step 3. Does it appear that exists? If so, what

appears to be the value of the limit when � is in radians?

5. In the diagram, AO � 1 and � is a small angle, in radians. Find the lengths of AB

and the arc AC of the circle centred at O. How is this evidence for the statement

?

�

A

CBO

limsin

�

�

�→=

01

limsin

�

�

�→ 0

limsin

�

�

�→ 0

sin �

�

00

limsin

.�

�

�→ 0

lim cos�

�→

=0

1lim sin�

�→

=0

0

1

–1

0

y

–� –�

2�

2�

y = sin�

�

–1

0

y

–�

2�

2�

y = cos�

�

1

8.38.3

8.3 Limits of Trigonometric Functions MHR 505

From the investigation, it seems that exists, and is

equal to about 0.017 if � is in degrees, and equal to 1 if � is inradians. The number 1 is a lot simpler to work with, and theresulting differentiation formulas are also much simpler, sowhen it comes to differentiating trigonometric functions, � is always measured in radians.On a graphing calculator, the limit also appears to be 1 when � is measured in radians. The screen shows the graph without the axes showing, so we can see the function at x � 0 better.

� 1 (valid when � is measured in radians)

Now that we know the three basictrigonometric limits,

and , we are

able to determine many others.

Example 1 Determining Another Important Trigonometric Limit

Determine .

SolutionSubstituting � � 0 into the expression fails, since the result, , is undefined. Before

determining the limit using algebra, we get a sense for what it might be by setting up tablesof values using the TABLE SETUP screen. Remember that we are using radians for theangle measure, as we anticipate that this might lead to a simpler result than using degrees.

Approaching 0 from the right: Approaching 0 from the left:

00

limcos

�

�

�→

−0

1

limsin

�

�

�→=

01lim cos ,

��

→=

01

lim sin ,�

�→

=0

0

limsin

�

�

�→ 0

limsin

�

�

�→ 0

The number 0.017 may seem

strange at first glance, but it

arises naturally—see question

10 in Section 8.4.

Window variables:x� [�2�, 2�], y� [�1.5, 1.5]Use the Format settings to turnthe axes off.

Web ConnectionFor an animated demonstration of

, go to

www.mcgrawhill.ca/links/CAF12 and follow the links.

limsin

x

xx→

=0

1

506 MHR Chapter 8

Remember that 1E-6 means 1 � 10�6 or 0.000 001.It seems that the limit is 0. Let us verify this with a convincing algebraic argument.

The limits and ��0 are used in determining the derivative

formulas for sine and cosine in Section 8.4. We continue to explore limits of trigonometricfunctions in this section.

Example 2 Determining More Trigonometric Limits

Determine each limit.

a) b) c)

a) Solution 1 Paper and Pencil MethodThe first method to try is to substitute x � 0 into the expression. This fails, since the numer-

ator and denominator are both 0 when x � 0. Instead, look for the basic limit, .

Solution 2 Graphing Calculator MethodWe use the TABLE SETUP screen to set up tables of values for x approaching 0 from theright and from the left.

lim sin lim sin

( )

x x

xx

xx→ →

=

=

=

0 0212

12

1

12

lim sinx

xx→ 0

lim tan sincosx

x xx x→

−0

3lim sinx

xx→ 0

2limsin

x

xx→ 0 2

limcos

�

�

�→

−0

1lim

sin�

�

�→=

01

limcos

limcos cos

cos

limcos

� �

�

�

�

�

�

�

��

→ →

→

− = − × ++

=

0 0

0

2

1 1 11

−−+

= −+ + =

→

11

11

0

22

� ��

� ��

�

(cos )

lim sin(cos )

cos ,Since sin the2 nn sin2cos .

limsin sin

(cos )

2

0

1

1

10

1 1

� �

�

�

�

��

− = −

= × −+

= × −+

→

WWe use the product law of limits.

= 0


The result from the graphing calculator agrees with the result in Solution 1:

.

b) Solution 1 Use the Double-Angle FormulaUsing the double-angle formula for sine, we obtain

c)

The strategy here is to express the function in terms of sine and cosine, simplify, and look

for the basic limit .

Example 3 Determining Another Trigonometric Limit

Determine .

SolutionSubstituting � � 0 into the expression produces , which is undefined. There are no algebraic

simplifications, so we suspect that the limit does not exist.

As � gets closer and closer to 0, the numerator gets closer and closer to 1, whereas thedenominator gets closer and closer to 0, so that the entire expression gets larger and largerin magnitude. Thus, the limit does not exist.

10

limcos

�

�

�→ 0

lim tan sincos

lim

sincos

sin

cos

lim

x x

x

x xx x

xx

x

x x→ →

→

− =−

=

0 0

0

3 3

33

30

sincos

sin

coscoscos

lim sin sin coscos

xx

x

x xxx

x x xxx

−×

= −→ 22

0 2

0 2

3

3

xx xx x

xx

xx

x

x

= −

= × −

→

→

lim sin ( cos )cos

lim sin coscos

== × −

=

1 3 11

2

2

lim sinx

xx→ 0

lim tan sincosx

x xx x→

−0

3

lim sin lim sin cos

lim sin cos

x x

x

xx

x xx

xx

x

→ →

→

=

=

= × ×

0 0

0

2 2

2

1 2 1

== 2

lim sinx

xx→

=0 2

12

Solution 2 Use First PrinciplesMatching the pattern for the basic limit,

, we get

lim sin lim sin

lim sin

x x

x

xx

xx

xx

→ →

→

= ×

=

0 0

2 0

2 2 22

2 22

(Note thhat as

= ×→ →

=2 1

0 2 0

2

x x, )

lim sinx

xx→ 0

508 MHR Chapter 8

As � approaches 0 through positive values, the expression is positive, whereas when �approaches 0 through negative values, the expression is negative. So, we can express moreprecisely that the limit does not exist as follows:

Examine the graph of the function on a graphing calculator. The calculator screen

is consistent with the conclusion that the limit does not exist. The function has aninfinite discontinuity at � � 0.

y = cos �

�

y = cos �

�

limcos

limcos

� �

�

�

�

� � ��

0 0� �

Key Concepts� Three fundamental limits of trigonometry are

Communicate Your Understanding1. Suggest two ways to use a graphing calculator to determine an approximate valuefor a limit.

2. In trying to determine a limit such as , why does substituting � � 0 not

work?

3. Describe the strategies used to evaluate trigonometric limits.

limsin

�

�

�→ 0

limsin

�

�

�→=

01lim cos

��

→=

01lim sin

��

→=

00

Practise1. Determine the value of each limit, if it exists.

a) b)

c) d)

e) f )

g) h)

2. Evaluate each limit, if it exists.

a) b)

c) d)

e) f )

g) h)

i) lim costanx

xx→

−0

1

lim tan sincosx

x xx x→

−0

5lim cosx

xx→

−0

2

2

1

lim cossinx

xx→

−0

1lim cosx

xx→

−0

1

limsin

x

xx→ 0

2

3

34

limsin

x

xx→ 0

2

2

34

limsin

x

xx→ 0

2 34

lim tanx

xx→ 0

52

lim sinx

xx→ 0

52lim sin

x

xx→ 0

34

lim sinx

xx→ 0

3lim sinx

xx→ 0 3

lim sinx

x→ 0 3lim sin

x

xx→ 1

lim sin( )x

xx

�0

1lim sinx

xx→ 0

B

Window variables:x� [�2�, 2�], y� [�3, 3]


Apply, Solve, Communicate3. Communication a) Use a calculator to

approximate the value of for

x � 0.1, x � 0.01, x � 0.001, and x � 0.0001.b) Use the results of part a) to estimate the

value of .

c) Use a calculator to approximate the value of

for x � 0.000 01, 0.000 001, and

0.000 000 1. Explain why this result is notconsistent with the result of part b).

4. Communication a) Evaluate .

b) Describe how you evaluated the limit in part a).

5. Application a) Evaluate .

b) Describe how you evaluated the limit in part a).

6. Thinking/Inquiry/Problem Solving Evaluate

.


8. Application Does exist? If so,

what is it? If not, why not?


10. Determine the value of each limit, if it exists.

a) b)

c)

11. Determine each limit.

a) b) c)

d) e) f ) lim tantanx

xx→ 0

34

limsintanx

xx→ 0

2

2

34

lim sintanx

xx→ 0

34

limsinsinx

xx→ 0

2 34

limsinsinx

xx→ 0

2

2

52

lim sinsinx

xx→ 0

52

limsin tan

sinx

xx→

( )0

lim sinx

xx

��

��

�

1lim cosx

xx→

−0 2

1

lim sin(cos )

cos

.x

x

x 0 1

lim sinx

xx 0

lim cos( ) cos .h

a h ah

� �0

lim sin( ) sinh

a h ah

� �0

lim cosx

xx

�0 2

2 1

lim tanx

xx 0

62

tan x xx

−3

lim tanx

x xx→

−0 3

tan x xx

−3

C

510 MHR Chapter 8

Derivatives of the Sine, Cosine, andTangent Functions

In this section, we develop the derivatives of the sine, cosine, and tangent functions.

Investigate & Inquire: Path of the Shadow of a BallMoving in a CircleA small ball is attached to a string 1 m long. The ball is swung aroundcounterclockwise in a vertical circle at a constant rate of one radian per second.

Sunlight from the right of the diagram shines on the ball and casts a shadow on a verticalscreen that is several metres to the left of the y-axis. The shadow’s motion on the screen isparallel to the y-axis and between y � �1 and y � 1. The sun’s rays are parallel to the x-axis,so the position of the shadow has the same y-value as that of the ball at every time.

When the ball is at the point (1, 0), a stopwatch starts timing the motion. Let t representthe time reading on the stopwatch and let (x, y) represent the position of the ball. Notethat t is measured in seconds, and x and y are measured in metres.

1. Let � be the angle between the string attaching the ball to the origin and the x-axis.Given that the rate at which the ball revolves is 1 rad/s, determine a relation between � and t.

2. Use the diagram to express the x- and y-coordinates of the position of the ball in termsof �.

3. Express the x- and y-coordinates of the position of the ball in terms of t.

4. In your notebook, sketch one cycle of the graph of the function y(t) obtained in Step 3.Leave room on your page for one of the same size under it that you will sketch in Step 9.Although y represents the position of the ball, in this investigation focus your attention onthe fact that y is also the position of the shadow. Thus, the graph of y(t) is the positionfunction of the shadow for the first 2� seconds of the motion. Conjecture a formula for y(t).

5. From the discussions about rates of change in Chapter 3, we know that the rate ofchange of the position of an object is its velocity. We can estimate the velocity of theshadow by finding the limits of average rates of change. For example, to estimate the

velocity v( ) of the shadow at t � , copy and complete the table.�6

�6

sun’s rays

y

x0

ballshadow

1–1

1

–1

�

8.48.4

8.4 Derivatives of the Sine, Cosine, and Tangent Functions MHR 511

6. From the table in step 5, estimate the velocity of the shadow at t � .

7. Using the same method as in steps 5 and 6, estimate the velocity of the shadow at the

following values of t: 0, , , , �, , and 2�.

8. Some of the velocities estimated in step 7 are positive and some are negative. Explainwhat the sign of the velocity means.

9. Using the data generated in steps 7 and 8, plot a graph of velocity vs. time for theshadow. Place it directly beneath the graph drawn in step 4, so that the corresponding t-values for each graph are aligned. In this way, the position of the shadow at a particulartime is given in the upper graph, and the velocity at the same time is given directly below it in the lower graph.

10. Conjecture a formula for the velocity function.

11. Is the velocity zero when the shadow is smallest, or when the shadow is largest? Explain.

12. Use the results of the investigation to conjecture a derivative formula for a trigonometricfunction.

Another way to discover the derivative function is with the aid of technology. Enter y � sin x on a graphing calculator as Y1. Now use the nDeriv function of the calculator asfollows. Graph the two functions.

It appears that the derivative of the sine function is the cosine function.

Conjecture:

The algebraic argument that follows verifies this conjecture, and makes use of the definitionof the derivative, the addition formula for sine, and the basic trigonometric limits.

ddx

x x(sin ) cos=

32��

2�3

�4

�6

�6

t �t y(t) �y � y(t + �t) − y(t) v = �

�

yt

0.2

�6

0.1

�6

0.05

�6

0.01

�6

0.001

�6

0.0001

y(t + �t)

Window variables:x� [�2�, 2�], y� [��, 1.5]

512 MHR Chapter 8

If f(x) � sin x, then, by the definition of the derivative,

Since and , as determined in Section 8.3,

f �(x) � sin x � 0 � cos x � (1)� cos x

Remember that, since the fundamental limit is valid only if the angle h is

measured in radians, the derivative formula just derived is valid only if the angles aremeasured in radians.

(for x measured in radians)

The derivative of the cosine function can be obtained using the Pythagorean trigonometricidentity and differentiating implicitly.

sin2 x � cos2 x � 1

Differentiating implicitly with respect to x and using the chain rule, we get

(for x measured in radians)

The derivative of the tangent function can be found by expressing tan x in terms of sin xand cos x and using the quotient rule to differentiate.

ddx

x x(cos ) sin= −

2 2 0

2 2

sin (sin ) cos (cos )

sin cos cos (cos )

x ddx

x x ddx

x

x x x ddx

x

� � � �

� � 00

22

ddx

x x xx

x

(cos ) sin coscos

sin

� �

� �

ddx

x x(sin ) cos=

lim sinh

hh→

=0

1

lim sinh

hh→

=0

1lim cosh

hh→

− =0

1 0

Expand using the addition formula for sine: f xh

�( ) lim (si� 0

nn cos cos sin ) sinx h x h xh

� �

Rearrange and look for basic limits:: f x x h x x hh

x hh

h

�( ) lim (sin cos sin ) cos sin

limsin (cos )

� � �

��

0

0

1 ��

� � ��

��

�

cos sin

lim sin cos cos sin

sin lim co

x hh

x hh

x hh

x

h

h

0

0

1

ss cos lim sinhh

x hhh

� �

10

f xf x h f x

hx h x

h

h

h

�( ) lim( ) ( )

limsin( ) sin

��

��

0

0

SInce h varies but x is fixed in the limiting

process, sin x and cos x can be factored to the

left of the limit signs.


Example 1 Derivatives of Trigonometric Functions

Differentiate.a) y � 2 sin (3x � 4) b) y � cos3 (4x2 � 7) c) y � 2 sin x cos x

Solution a) Using the chain rule, we have

b) Using the chain rule repeatedly, we get

c) Solution 1 Use the Double-Angle FormulaRecognizing the pattern of the double-angle formula, we havey � 2 sin x cos x� sin 2x

Using the chain rule, we gety� � (cos 2x)(2)� 2 cos 2x

dydx

xddx

x

x x

= − −

= − − −

3 4 7 4 7

3 4 7 4 7

2 2 2

2 2 2

cos ( ) [cos( )]

cos ( )[ sin( )] dddx

x

x x x

x x

( )

cos ( )[ sin( )]( )

cos ( ) s

4 7

3 4 7 4 7 8

24 4 7

2

2 2 2

2 2

−

= − − −= − − iin( )4 72x −

dydx

x ddx

x

x

� � �

� �

2 3 4 3 4

6 3 4

cos( ) ( )

cos( )

ddx

x x(tan ) sec= 2

y x

xx

dydx

x ddx

x x ddx

x

x

x

�

�

��

�

tan

sincos

cos (sin ) sin (cos )

cos

cos (

2

ccos ) sin ( sin )cos

cos sincos

cos

sec

x x xx

x xx

x

x

� �

� �

�

�

2

2 2

2

2

2

1

Web ConnectionFor an animated development of the trigonometric derivatives, go towww.mcgrawhill.ca/links/CAF12 and follow the links.

514 MHR Chapter 8

Solution 2 Use the Product RuleUsing the product rule with f (x) � 2 sin x and g(x) � cos x, we gety � f (x) g (x)y� � g(x) f �(x) � f (x) g�(x)� cos x (2 cos x) � 2 sin x (�sin x)� 2 (cos2 x � sin2 x)� 2 cos 2x

Example 2 Derivatives of More Complicated Trigonometric Functions

a) Show that .

b) Show that if , then .

Solutiona) Use the chain rule on the composite functions.Let y � 3 ln(tan x) ��esin2 x.

b) Solution 1 Paper and Pencil MethodUsing the quotient rule with f (x) � x and g (x) � tan x, we get

Rewriting as two fractions, and expressing in terms of sine and cosine, we get

y xx

x xx

xx x

xx

� � � ��

��

� �

�

�

tantan

sectan

tancossincos

2

2

2

2

2

2

11

�

�

��

� � ��

��

� �

cotsin

cot csc

x xx

x x x

12

2

y xx

ygf fg

gx x x

xx x x

x

�

��

� �

� �

tan

tan ( ) (sec )tan

tan sectan

��

2

2

2

2

2

1

dydx x

ddx

x eddx

x

xx e

x

x

= −

= −

3

3

2

2

2

2

tan(tan ) (sin )

tan(sec ) ( )

sin

sin 22

3 2

3

2

2

2

sin (sin )

sectan

( ) sin cos

sectan

sin

s

xddx

x

xx

e x x

xx

e

x= −

= − iin sin2

2x x

y x x x′ = −cot csc2yx

x=

tan

ddx

x ex

xe xx x( ln(tan ) ) sec

tansinsin sin3 3 2

2 22

− = −


Solution 2 Computer Algebra MethodWe use a computer algebra system to find the derivative.

First we determine the derivative, and then store it in variable a. Since the solution is notyet in the format asked for, we expand it. We can see that the first term is equal to cot xand the second term is equal to x csc2 x. Thus,

y� � cot x � x csc2 x

Example 3 Finding Tangents and Local Extrema

For the function f (x) � 4x � tan x in the interval [� , ], find

a) an equation of the tangent at the point where x �

b) the local extrema

Solutiona) f (x) � 4x � tan xf �(x) � 4 � sec2x

The slope of the tangent is .f � �4

��

��

�4

�2

�2

516 MHR Chapter 8

The point of tangency is

The equation of the tangent, using the point-slope method, is

b) Determine the x-values for which the derivative is zero or does not exist. The derivativeis f �(x) � 4 � sec2 x.The x-values for which sec2 x does not exist are � and , but these are asymptotes, not local

extrema. Thus, we need determine only the x-values for which the derivative is zero. Setting thederivative equal to zero and solving for x, we get

4 � sec2 x � 0sec2 x � 4

cos2 x �

cos x � or cos x � �

x � or �

To test the critical numbers, we use the second derivative test. f �(x) � 4 � sec2 x

� 4 � (cos x)�2

The second derivative is

f x x x

xx

�( ) (cos ) ( sin )

sincos

� �

� �

�2

2

3

3

�3

�3

12

12

14

�2

�2

y x

y x

� � � ��

��

� � �

( )��

�

1 24

22

1

� � � � �

��

4 4 44

4 4

41

, , tan

,

f

=

−

= −

f � � �

�

44

4

4 1

4

4 1

12

2

2

2

2

��

��

�

��

� ��

��

� ��

��

�

sec

cos

Windowvariables:

x� [� , ],y� [��, 4]

�2

�2


Substituting the critical numbers into the formula for the second derivative, we get

There is a local maximum at x � . There is a local minimum at x � � .

The y-coordinates of the local extrema are as follows:

There is a local maximum at and a local minimum at .− − +

� �3

43

3,� �3

43

3, −

f −

= −

−

= − +

� � �

�

34

3 3

43

3

tanf� � �

�

34

3 3

43

3

=

−

= −

tan

�3

�3

f � ��

��

� ��

��

��

��

��

��

�

�

�

�3

23

3

2 32

12

8 3 0

3

3

sin

cos

f � ��

�3

23

3

2 32

12

8 3 0

3

3

��

��

�

��

��

��

� � !

sin

cos

Key Concepts� The trigonometric derivative formulas are valid only for radian measure.

Communicate Your Understanding1. Why is radian measure used exclusively when dealing with the derivatives oftrigonometric functions?2. Which of the three functions y � sin3 x, y � sin x3 and y � (sin x)3 are identical?How would you differentiate each of these functions?3. a) Sketch the graph of the cosine function in the domain x� [��, �]. Directlybelow it, sketch the graph of the sine function. Make sure to use the same scale foreach graph and to align the y-axes of both graphs.b) Sketch some tangents on the cosine graph. Measure the slopes of the tangents, andcompare the slopes with the corresponding heights of the sine graph (directly below).What conclusion does this support?4. Given the derivative of the sine function, how is the derivative of the cosinefunction developed? How is the derivative of the tangent function developed?5. In Example 3 above, the function f (x) � 4x � tan x is odd. Explain how this factcan be used to provide a partial verification of the calculations.

ddx

x xddx

x xddx

x xsin cos cos sin tan sec= = − = 2

518 MHR Chapter 8

Practise1. Determine the derivative of y with respect tox for each function.a) y � sin (4x � 7) b) y � sin (x2 � 3)c) y � cos (5x � 3) d) y � 3 tan (2x � 3)e) y � cos (3x2 � 1) f ) y � tan (5x2 � 1)g) y � sin (cos2 x) h) y � sin (cos x)

2. Differentiate.

a) b) y � x sin (3x � 2)

c) y � x2 sin (2x2 � 5) d)

e) g(x) � x cos (8x � 17) f ) y � x2 cos (0.4x2 � 9)g) y � 5x2 tan3 (3x2 � 1) h) h(x) � 4x2 sin3 (6x2 � 2)

3. Find the derivative.a) y � sin3 (cos2 x) b) f (x) � cos4 (sin x3)c) y � x cos (tan x) d) y � sin (ex)e) h(x) � cos (ln x) f ) g(x) � ln (sin (ex))

g) y � sin2 x � cos2 x h)

Apply, Solve, Communicate 4. Determine the equation of the tangent to thecurve at the given point.

a) y � x sin 2x, ( , )b) y � cos2 x, ( , )c) y � , ( , 1)5. Determine any extrema.a) y � cos x � sin x, x �[��, �]b) y � sin2 x � sin x, x �[��, �]

c) y � 2cos x � cos 2x, x �[ , ]d) y � � tan x, x �[� , ]6. Determine the points of inflection.a) y � 2 cos x � sin 2x, x�[0, �]b) y � 2sin2 x � 1, x�[��, �]

c) y � sin x � tan x, x�[� , ]

7. Communication/Inquiry/Problem Solving Beginwith the function f (x) � sin x. a) Determine the derivative of f. Then,determine the second derivative of f. Continueto find the first eight derivatives of f. Report anypattern that you notice.b) What is the 87th derivative of f? What is the138th derivative of f?c) Develop a formula for the nth derivative ofthe sine function.d) Develop a formula for the nth derivative ofthe cosine function.

8. Application The equation y� � k2y � 0 isknown as a differential equation, since it involvesa function y and its derivatives. This equation iswidely used in applications to model physicalsystems that oscillate. For example, y couldrepresent the voltage in an electrical circuit, orthe position of a vibrating airplane wing.

a) Show that the function y � A sin kx �B cos kx satisfies the differential equation for allvalues of the constants A, B, and k.b) Show that the function y � C sin (kx � D)satisfies the differential equation for all values ofthe constants C, D, and k.

9. A line passes through the point

and is tangent to the graph of y � sin x at a

point for which x� [0, ]. Determine the

equation of the line.

10. Inquiry/Problem Solving This question explainswhere the number 0.017" comes from when

we use degrees to determine . (See the

investigation in Section 8.3.) Let the angle A be

limsin

�

�

�→ +0

�2

− −

3

6, �

32��

2

�2

�2

1cos x

32��

2

�4

12tan x

14

�3

�4

�4

f xx

x( ) sin

cos= +1

f xx

x( ) cos=

f xx

x( ) tan=

B

C


Application

Communication

Thinking/Inquiry/Problem Solving

Knowledge/UnderstandingA baseball hit at an angle � to the horizontal, with initial velocity

v0, has horizontal range, R, in metres, given by .

The constant g � 9.8 m/s2 represents theacceleration due to gravity.

a) Sketch R for an appropriate

value of v0 for � � [0, ]. Is this domain

appropriate? Explain.b) What angle gives the maximum range? c) What is the maximum range?

�2

Rvg

= 02

2sin ( )�

Achievement CheckAchievement Check

MODELLING

•

M AT H

•

measured in degrees. Define � to be the sameangle as A, but measured in radians.

a) Express � in terms of A. Determine .

b) Let y � sin �. Using the chain rule,

and the result of part a), determine . The

result is the derivative formula for the sinefunction that is valid when the angle ismeasured in degrees. c) Compare the formula obtained in part b) to the result obtained in Step 1 of the investigation in Section 8.3. Is the formulaconsistent with this result? What is the exactvalue of the number 0.017"?d) Determine the derivative formula for thecosine function if the angle is measured in degrees.

11. a) Sketch the graph of the function y � sin #x #. For which values of x is the functionnot differentiable?b) Repeat part a) for the function y � #sin x #.

12. Determine the values of x for which the

derivative of the function f (x) � cos (x � ) is 0.

13. Determine .

14. a) Find a function whose derivative istan2 x.b) Find a function whose derivative is tan x.c) Find a function whose derivative is sec x.

limtan

h

h

h→

+

−

0

41�

1x

dydA

ddA

�

The word trigonometry comes from the Greek words for triangle, trigon, and measurement,metron. Geometry was taken very seriously by the ancient Greeks, and Archimedes(287–212 B.C.) is said to have died as a result of concentrating so hard on a geometric problemthat he failed to notice that his home city of Syracuse was under Roman attack. The soldier atwhom Archimedes snapped, “Do not disturb my circles,” lost his temper and killed themathematician on the spot.

Historical Bite: Geometry in Ancient Greece

520 MHR Chapter 8

Technology ExtensionUsing the TI-92 for Trigonometry

A computer algebra system such as the TI-92can help keep track of all the relationships usedin trigonometry.

The double-angle formulas for sine and cosine are derived from the addition formulas. Examples are shown in this screen, using the tExpand function.

This screen demonstrates some basicrelationships that can be derived from theaddition and subtraction formulas for sine and cosine. The last is a numerical example of the relationship cos (2� – �), where

� � .

Trigonometric equations can be solved as well, but some care must be taken in interpret-ing solutions. In this example, a basic equation involving the cosine functionis solved using the solve function on theAlgebra menu. The result represents two setsof solutions.

The first set of solutions should be interpreted

as and . The

ampersand and the digit 1 on either side of thevariable n indicate that these solutions should beinterpreted as all possible solutions in thedomain � � R. All of the solutions can beobtained by substituting integer values for n.The digit 1 indicates that this is the firsttrigonometric equation that has been solved inthis work session. Similarly, the digit 2 in thesecond set of solutions indicates that this is thesecond equation that has been solved. Thesecond set of solutions should be interpreted as

and � � 2n� � �, or � � (2n � 1)�.��= −( )4 1

2n

��= −( )12 5

6n

��= +( )12 5

6n

�6

G R A P H I N G C A L C U L A T O RG R A P H I N G C A L C U L A T O R



Technology Extension: Using the TI-92 for Trigonometry MHR 521

Trigonometric limits can be evaluated easilyusing a computer algebra system. The two basic limits for trigonometry are shown inthe screen below, using the limit function on theCalculus menu.

Finally, trigonometric derivatives are shown inthis screen. Note how the chain rule has beenused to evaluate the derivatives in the secondand third expressions, using the differentiatefunction on the Calculus menu.

Practise1. Use a computer algebra system to reduceeach expression to a simpler forma) cos (��) b) sin (� � �)c) tan (� � �) d) sec (2� � �)e) sin (� � �) f ) csc (2� � �)g) tan (� � �) h) cos ( � �)2. Determine whether each function is even orodd.a) f (x) � sin � b) f (x) � cos �c) y � tan �

3. Solve each equation.a) 4 sin2

� � 1 � 0b)

c)d) 2 sin 2� (cos � � sin �) � 2 cos 2�

� sin � � cos �

4. Evaluate using a computer algebra system.

a) b)

c) d)

e) f)

5. Differentiate.a) cos2x b) sin x3

c) x3 tan 2x d) sin2ex

e) cos 2x sin 3x f )sincos

23xx

lim sincosh

hh� �� 1

lim tansinx

xx→ 0

53

lim sinx

xx→ +1

2lim sinx

xx� �

1

lim sinx

axx→ 0

lim sinh

hh→ 0

2

3 3 1 12tan ( ) tan� �+ − =2 2 2 2 3 3sin sin cos� � �+ = +

�2



522 MHR Chapter 8

The periodic behaviour of sine and cosine functions, which are collectively known assinusoidal functions, makes them well suited for modelling many of the rhythmicphenomena of our world. The rotation of Earth, the motions of the planets around the sun,the seasons, the tides, even some of the rhythms of our bodies are sinusoidal, or nearly so.Trigonometric functions are also used extensively to model oscillating systems and themotion of all kinds of waves: everything from electric circuits to vibrating machinery, and including descriptions of light, sound, music, the distribution of electrons within anatom, and many other situations.

Tidal forces on Earth are due to the gravitational pull of thesun and the moon on Earth. In this investigation, you willdetermine a formula that models the tides, and then use thatformula to calculate the time between successive peak tides.You will use a very simplified model. The actual situation isquite complicated, and depends on many factors, such asthe shape of nearby coastlines, the shape of the ocean floor,and the size of the body of water. For example, in most placeson Earth there are about two high tides per day, but there isonly one high tide per day in the region of Tonkin in Vietnam.Also, the times at which the high tides occur vary from day today almost everywhere on Earth, but not in Tahiti, an islandin the Pacific Ocean, where the high tides occur at the sametime every day!

Investigate & Inquire: Modelling the Tides1. Consider the sun to be fixed. The sun causes the water on Earth to “bulge” in twoplaces: toward the sun at the point on Earth facing the sun, and away from the sun at thepoint on Earth facing away from the sun. To simplify the calculations, assume that thebulges do not move, and that Earth just rotates through them.

Then, a specific point on Earth will encounter a water bulge every 12 h, as Earth rotates.Assuming that the ocean level is a sinusoidal function of time, write a function of the form y � sin kt, where t is time, in hours, to model the ocean level experienced at a specific point onEarth. The period should be 12 h, since that is the time between peaks. Determine the valueof k. (Note that we are assuming that the amplitude of the tides caused by the sun is 1 unit.)

Sun

Earth

high tide

Modelling With Trigonometric Functions

MODELLING

•

M AT H

•

8.58.5

8.5 Modelling With Trigonometric Functions MHR 523

2. Imagine that the moon is overhead right now. Since it orbits Earth, moving in the samedirection that Earth rotates, it will take more than 24 h for the moon to be directlyoverhead again. (When 24 h have passed, the moon will have moved along its orbit a bit, soit will take a little more time before Earth catches up again so that the moon is overhead.)

Determine how long it takes for the moon to be overhead again. (It takes 27.3 days for themoon to go around Earth once.)

3. Use the period you found in step 2 to write a sine function that models the tidesdue to the moon. The amplitude should be 2.2 times greater than the amplitude in theformula for the tides due to the sun. (Thisratio will be worked out in Example 1.)Thus, the function should be of the form y � 2.2 sin mt. Determine the value of m.

4. Add the two functions found in steps 1 and3 to obtain a model for the tides. Sketch agraph of the tide-model function over a periodof several days.

5. Judging from the graph, is the time interval between high tides roughly constant? Whatis the period? Does this agree with the observation that there are roughly two high tidesper day at most places on Earth?6. The following data are from Burncoat Head, Nova Scotia, on the Bay of Fundy, whichholds the world record for the highest tide. Compare these data with your model byplotting the data points on a scatterplot and superimposing your model on the plot.

7. Use SinReg on the STAT CALC menu of a graphing calculator to fit a trigonometricregression curve to the data of step 6. Compare the regression curve with the model fromstep 4. How accurate is the model?

Earth

t = 0

moon

Earth

t = 24 h

moon

Height (m)

Hour

11.8

12

11.1

13

9.2

14

6.5

15

3.9

16

1.8

17

0.8

18

0.9

19

2.3

20

4.9

21

7.8

22

10.3

23

Height (m)

Hour

11.6

00

10.4

01

8.1

02

5.5

03

3.0

04

1.4

05

0.8

06

1.5

07

3.4

08

6.2

09

8.9

10

11.0

11

Web ConnectionGo towww.mcgrawhill.ca/links/CAF12to obtain daily tide data.

Windowvariables:x � [0, 24], y � [0, 12]

524 MHR Chapter 8

In step 1 of the investigation, it was assumed that water bulges caused by the sun do notmove. However, they will move, since Earth moves around the sun, completing one circuitof its orbit per year. You might like to explore how to modify your formula from theinvestigation to take this into account.

As mentioned previously, tidal forces on Earth are due to the gravitational pull of the sunand the moon on Earth. More precisely, tidal forces are due to the fact that gravitationalforces decrease with distance. Because the sun’s pull on the side of Earth facing it is greaterthan the sun’s pull on the side of Earth facing away from it (since it is farther away), Earthtends to be stretched. The moon’s pull has the same effect as the sun, and when thedirections of the two stretches line up, as when there is a full moon or a new moon, thenthe tides are especially high.

As an approximation, we can imagine that the water bulges stay in roughly the same placeas Earth rotates through them. Since there is a water bulge on each side of Earth, anyparticular place on Earth encounters a bulge (high tide) twice per day in thisapproximation. (In reality, high tides are about 12.5 h apart.)

Tidal forces act on the solid part of Earth as well as on the water. As municipal publicworks departments are well aware, there are more water-main breaks at full moon andnew moon than at other times of the month!

Example 1 Comparison of the Tidal Forces Due to the Sun and the Moon

The tidal force is the rate of change of the gravitational force.a) Determine the tidal forces on Earth due to the sun and the moon.b) Compare the two forces.

We need the following data.Mass of sun: 2.0 � 1030 kgMass of moon: 7.4 � 1022 kgMass of Earth: 5.98 � 1024 kgDistance between sun and Earth: 1.5 � 1011 mDistance between moon and Earth: 3.84 � 108 mG � 6.67 � 10�11 N m2/kg2

Solutiona) The function that describes the gravitational force of the sun on Earth is

where F is the force, G is a constant that makes the units come out conveniently, M isthe mass of the sun, m is the mass of Earth, and x is the distance between the sun andEarth.

To determine the tidal force acting on Earth, we differentiate F with respect to x:

For the gravitational force of the moon on Earth, the same formula works provided that M is the mass of the moon, and x is the distance between Earth and the moon.

F x G Mmx

�( ) � �2 3

F x GMmx

( ) = 2

The unit of force used

here is the newton, N.


Hence, the tidal force acting on Earth due to the moon is

b) To compare the relative strengths of the tidal forces, we can ignore the minus sign,since it tells us only about the direction of the forces. Substituting the values for the sunand Earth gives

Similarly, substituting the values for the moon and Earth gives

The ratio of the two expressions (without simplifying first) is about 2.2. Thus, the tidalforce exerted by the moon is more than twice as big as the tidal force exerted by the sun.

Is the result of the preceding example surprising? The fact that the tidal force of the moonon Earth is greater than that of the sun on Earth is interesting, especially in light of the factthat the gravitational force of the moon on Earth is much smaller than the gravitationalforce of the sun on Earth. You can determine the ratio of the gravitational forces inquestion 1 on page 529.

If the sun, Earth, and the moon did not move relative to each other, and Earth did notrotate, then the tides would be constant. However, they do move, and so the tides go inand out.

An object hung from a car’s rearview mirror starts to swing when the car picks up speed orslows down, but stays steady if the car has been going at a constant speed in the samedirection for a while. This motion is the subject of the following example.

Example 2 Pendulum Swinging in a Moving Car

a) A car accelerates in a straight line at 1.4 m/s2. Determine the angle � that a pendulumhanging from the rearview mirror makes with respect to a vertical line.b) If the acceleration of the car begins to increase at the rate of 0.2 m/s3, atwhat rate does the pendulum’s angle increase?

Solutiona) Consider the forces acting on the pendulum bob. Gravity tries to pullthe bob (marked B in the diagram) down with a force equal to the weightmg of the bob. (The mass, in kilograms, of the bob is m, and g � 9.8 m/s2 isa constant that describes the strength of Earth’s gravity.)

For simplicity, we assume that the only other force acting on the bob is the tension, T, inthe rope. The tension can be broken up into two components. The vertical component,

Tidal force of the moon on Earth ��

�2 6 67 10 7 4 10 511 22( . )( . )( .. )( . )

.

98 103 84 10

1 0 10

24

8 3

12

�

�

� �

Tidal force of the sun on Earth ��

�2 6 67 10 2 0 10 511 30( . )( . )( .998 101 5 10

4 7 10

24

11 3

11

�

�

� �

)( . )

.

F x GM m

x�( ) moon

moon

� �23

B

mg

T cos�

T sin�

T

�

�

526 MHR Chapter 8

T cos �, must balance the weight exactly (since the pendulum does not rise or fall).T cos � � mgThe horizontal component, T sin �, provides the acceleration that forces the pendulum tokeep up with the car. According to Newton’s second law of motion, the force T sin � isequal to the mass of the bob times its acceleration, a. Putting all this together, we getT sin � � maDividing the second equation by the first results in

tan � �

Substituting the given data gives us

tan � �

�

From the diagram of the pendulum, we can see that � � [0, ].

Thus, � � 0.1419 rad (or approximately 8°).

b) We need to find . Differentiating tan � � implicitly with respect to time, we get

Substituting g � 9.8, � � 0.1419, and , we get

The angle changes at a rate of approximately 0.02 rad/s.Converting to degrees, we obtain

The angle changes at a rate of approximately 1.15/s.

ddt�

�=

0 02 180

1 15

.

.�

ddt� =

19 8 0 1419

0 2

0 02

2. sec .( . )

.�

dadt

= 0 2.

tan

sec

se

�

��

�

�

�

�

ag

ddt g

dadt

g

ddt g

2 1

1

Note that is a constant.

cc2�

dadt

ag

ddt�

�2

17

1 49 8..

ag

Window variables:

x � [0, ], y � [�1, 1]�2


In reality, accelerations are rarely smooth. If you try this yourself (while someone else isdriving!), the pendulum will probably start swinging wildly after a short time. Note that theresults of the example do not depend on the length of the pendulum.

Example 3 Maximum Area of a Triangle Inscribed in a Quarter-Circle

In computer geometry programs, points may be animated along an arc of a circle. In thefigures, the point A on a quarter-circle of radius can be dragged or animated alongthe arc. The inscribed right triangle changes as A moves along the arc. What is themaximum area of the right triangle?

Solution 1 Use Calculus

Substituting this value into the formula for A gives the maximum area.A � 50 sin 2�

� 50 sin 2( )� 50

�4

10 2 cosz �

10 2 sinz �10 2z

�

point A(drag)

10 2z 10 2z

point A(drag)

10 2z

point A(drag)

10 2

We let � represent one of the acute angles inthe triangle. Then, the unknown sides of the

triangle are sin � and cos �. Thearea of the triangle is

A � ( ) ( sin �)( cos �)

� 100 sin � cos �� 50 sin 2� (double-angle formula)

The maximum area occurs when � 0.

� 100 cos 2�

0 � 100 cos 2�0 � cos 2�

Since � is acute, � � . Thus, 2� � [0, �].

22

4

��

��

�

�

02

, �

dAd�

dAd�

10 210 212

10 210 2

528 MHR Chapter 8

We verify that this is a maximum using the second derivative test.A � �200 sin 2�

A( ) � �200 sin

� �200 � 0Thus, � � gives the maximum area of 50.

Solution 2 Use Trigonometry

As in Solution 1, the area of the triangle is A � 50 sin 2�.

The maximum value of any sine function is equal to its amplitude. Thus, the maximumvalue of the area of the triangle is 50.

Example 4 Teaching a Child to Catch

Grace is tossing a ball to her young nephew, who insists onstanding 3 m away. At what angle should she throw the ballso that it arrives with minimum speed?

The path of the ball is given by

where � is the angle from the horizontal that the ball is tossed, g is the acceleration due togravity, v0 is the initial speed with which the ball is thrown, x is the horizontal position ofthe ball, and y is the vertical position. Assume that the ball is caught at the same heightfrom which it is thrown, and label this height y � 0. All distances are in metres.

SolutionWe need an expression for the final speed, v, in terms of �. The path of the ball, y � f (x),is a quadratic function, with vertex halfway between Grace and her nephew. Bysymmetry, the final speed v equals the initial speed v0, provided that there is no airresistance. Thus,

We know that at the arrival point (the nephew), y � 0 and x � 3, so we can substitute thesevalues and solve for v.

0 33

2

92

3

32

32

2

2 2

2 2

2

� �

�

�

�

tancos

costan

sin cos

sin

��

��

� �

�

gv

gv

vg

g

y xg

vx= −

(tan )cos

��2 2 2

2

y xg

vx= −

(tan )cos

��2 0

2 22

�4

�2

�4

ball

pathy

x

�


We can find the minimum speed using calculus, as in Solution 1 of Example 3 on page 527,but it is often simpler in problems like this to use basic trigonometry.

The minimum speed occurs when sin 2� is a maximum. The maximum value for any sinefunction is equal to its amplitude. Thus, the minimum speed occurs when sin 2� � 1.

The minimum catching speed for her nephew occurs when Grace throws the ball at anangle of radians.�

4

sin 2 1

22

4

�

��

��

�

�

�

��

��

2 32

32

�

�

g

g

sin

sin

Apply, Solve, Communicate

1. Application The gravitational force between abody and Earth is given by

where F is the force, G is a constant, M is themass of the body, m is the mass of Earth, andx is the distance between the body and Earth.The following data are relevant:Mass of the sun: 2.0 � 1030 kgMass of the moon: 7.4 � 1022 kg

Mass of Earth: 5.98 � 1024 kgDistance between sun and Earth: 1.5 � 1011 mDistance between moon and Earth: 3.84 � 108 mG � 6.67 � 10�11 N m2/kg2

a) Determine the gravitational force that thesun exerts on Earth.b) Determine the gravitational force that themoon exerts on Earth.c) Determine the ratio of the quantity in part a)to the quantity in part b).d) Compare the result of part c) to the result ofExample 1 (pages 524–525).

F x GMmx

( ) = 2

B

Key Concepts� Sine and cosine functions are also called sinusoidal functions.� Periodic behaviour can often be modelled by sinusoidal functions.

Communicate Your Understanding1. What is a sinusoidal function? For what kinds of phenomena are sinusoidalfunctions useful models?2. Most of the applications studied in this section have models based on sine or cosinefunctions. Why do you think the tangent function did not appear very much?3. For a swinging pendulum, explain what angular speed means.

530 MHR Chapter 8

2. Inquiry/Problem Solving Consider the tidalforces acting on a futuristic spacecraft with mass104 kg, at the surface of a star. Assume that thespacecraft has a shield that can withstand theintense heat at the surface of the star.a) How great would the sun’s tidal force on thespacecraft be if it were at the sun’s surface? (The radius of the sun is about 7 � 108 m.)b) Repeat part a) for a white dwarf star thathas the same mass as the sun, but has a radiusof only 106 m.c) Repeat part a) for a neutron star that has thesame mass as the sun, but has a radius of only104 m.

3. Communication A right triangle has ahypotenuse of 10 cm.a) To determine the maximum area of such atriangle, let x represent the length of one of theunknown sides of the triangle. Use thePythagorean theorem to determine a formula, interms of x, for the length of the other unknownside of the triangle. Then, determine theexpression for the area of the triangle and findits maximum value.b) Compare this method of solution to themethod of Example 3. Which do you prefer?Explain.

4. Application Two sides of a triangle havelengths 15 m and 20 m. The angle between them

is increasing at a rate of rad/s. At what rate

is the length of the third side changing when

the angle between the other two sides is ?

5. The position, x, of the midpoint of avibrating violin string is described by theformula x � 0.05 cos (880 � t), where x ismeasured in centimetres and t is measuredin seconds.a) Determine formulas for the velocityand acceleration of the midpoint of the string.b) Determine the maximum speed of vibrationof the string, in centimetres per second.c) Show that the acceleration and positionsatisfy the differential equation

� (880�)2 x � 0.

6. A piston in a car engine moves up and downbecause its shaft is attached to a crank thatmoves in a circle at a constant rate, as shown inthe diagram.

a) Determine a formula for h in terms of �.b) Determine a formula for the velocity of thepiston in terms of the rotational velocity of the

crank, .

c) For which angles is the piston speed zero?d) For which angles is the piston speed amaximum?

7. Inquiry/Problem Solving The electricity flowingthrough the wires in your home is alternatingcurrent (AC). The word “alternating” meansthat the electricity changes direction. A simplifiedmodel of the current is V � 120 cos 120� t,where V is the voltage, in volts, and t is thetime, in seconds.a) How many cycles does the electricity makeper second? (That is, what is the frequency ofoscillation?)b) How many times does the electricity changedirection per second?c) The force that pushes the electricity throughthe wires is called the electric force field, orelectric field for short. The value of the electricfield is the negative of the derivative of thevoltage. Determine a formula for the electric field.d) When is the electric field at a maximum? What is the value of the voltage at those times?e) When is the electric field zero? What is thevalue of the voltage at those times?

8. Determine the maximum perimeter of a right triangle with hypotenuse cm.10 2

ddt�

1 2–1–2

1

2

3

4

–1

0 x

y

4

(0, )h

�

d xdt

2

2

�3

�90

2


9. Application A weather balloon rises straightup in the air, starting from the ground at pointA. An observer located at point B, 1 km from A,is tracking the balloon with a telescope. Theangle between the telescope’s line of sight andthe ground is �. The balloon rises at a constantspeed of 10 km/h.a) Determine a formula for � in terms of time t.b) Determine the rate at which � is changing

when � � .

10. An isosceles triangle is inscribed in a circleof radius r. Determine the angle � for which thearea of the triangle is a maximum.

11. Inquiry/Problem Solving a) A projectile thrownover level ground, at an angle � to the ground,

has a range R given by R � ( ) sin 2�, where v

is the initial speed, in metres per second, andg � 9.8 m/s2. Determine the angle of projection �for which the range is a maximum.

b) If the projectile is thrown over a slope, then

the range is , where

� is the angle that the slope makes with the horizontal.

Determine an expression for the angle �, interms of the angle �, for which the range is amaximum.

c) Test the result of part b) by substituting theappropriate value of � for the special case ofpart a). Do you obtain the same result as in parta) in this way?

12. A pendulum is swinging back and forth.The angle � that the pendulum’s string makeswith a vertical line can be modelled by thefunction � � 0.15 cos 6 t.Determine the maximum angular speed

and the maximum angular acceleration of the pendulum.

13. A rising weather balloon is being trackedwith a telescope by an observer 1 km away fromthe point at which the balloon was launched.The balloon has a component of velocity of10 km/h upward, and a light breeze causes theballoon to have a horizontal component ofvelocity of 5 km/h, directly away from theobserver. The angle between the telescope’s lineof sight and the ground is �. a) Determine a formula for � in terms of time t.b) Determine the rate at which � is changing

when � � .

14. Communication An isosceles triangle iscircumscribed about a circle of radius r.a) Determine the angle � between the twoequal sides of the triangle for which the area ofthe triangle is a minimum.b) Determine the angle � for which theperimeter of the triangle is a minimum.c) Compare the results of parts a) and b). Didyou expect them to be the same? Explain.

15. A hockey player is rushing downthe ice on a path parallel to theboards and 2 m wide of the net. Ifthe width of the net is 2 m, at what point should the player releasethe puck to maximize the shot angleto the net? (The shot angle is theangle formed by the puck and thetwo goalposts.)

�6

ddt

2

2

�

ddt

�

� �R

Rvg

= −

2 2

2

sin ( ) coscos� � �

�

�

R

vg

2

�

�6

B

�

net2 m

2 m

532 MHR Chapter 8

Application

Communication


Knowledge/Understanding


Sunlight (and most light) has a mixture of colours in it. However,laser light is nearly monochromatic, that is, it contains only onecolour. A certain beam of monochromatic light can be described bythe function E � A cos (kx � wt), where E is the electric field in thelight beam, A is the amplitude of the electric field, x is the positionalong the beam of light, t is time, and k and w are constants thatcharacterize the light. Notice that E depends on two variables, xand t.a) If we could take a “photograph” of the profile of a light beam,the profile would look sinusoidal. This amounts to substituting aparticular time into the formula for E. For example, we couldsubstitute t � 0 to get E � A cos (kx). The distance between peaksof this profile is known as the wavelength of the light. If thewavelength of this light is 6 � 10�7 m, determine the value of k.b) Sketch a graph of the function of part a) for two periods ofoscillation.c) Instead of taking a “photograph” of the profile, we could focuson a particular location on the x-axis (such as x � 0), and measureE as the light wave passes by. Again we would get a sinusoidal vari-ation in the value of the electric field E � A cos (wt). The distancebetween peaks of this graph is known as the period of the light, andthe reciprocal of the period is known as the frequency of the light.If the frequency of this light beam is 5 � 1014 oscillations persecond, determine the period. Also determine the value of w, whichis known as the angular frequency.d) Sketch the graph of the function of part c) for two periods ofoscillation.e) Do some research on light to find out what colour of light isbeing dealt with in this exercise. Report on and justify yourfindings.f ) Determine the product of the wavelength and the frequency ofthis light beam. What are the units of this quantity? What does thisquantity have to do with the light beam? g) Is the light beam travelling to the left or to the right along thex-axis?

Investigate & Apply: Modelling Daylight Hours MHR 533

In Leamington, Ontario, the number of daylight hours in a day, S, is about 15 hon the longest day of the year, and about 9 h on the shortest day of the year,where S represents the length of time between sunrise and sunset.

How well can this phenomenon be modelled using trigonometric functions? Usethe following questions to guide your investigation, and create a summary report.

1. Construct a formula for S, assuming that it can be modelled by a sinusoidalfunction of time.

2. On which days of the year is the magnitude of the rate of change of S amaximum?

3. How does the fact that a year is a little longer than 365 days affect the validityof the model in the long run?

4. Use data for sunrise and sunsetto test the model that you haveconstructed. How good is yourmodel?

5. How is the amplitude of themodelling function affected by achange in latitude? (Note that, abovethe Arctic circle, the longest day of theyear is 24 h long.) Modify your formula to take latitude into account. Test yourformula using sunrise and sunset data for places with various latitudes. How goodis your model?

6. Are there any other factors that would have an effect on the formula?

MODELLING

•

M AT H

•

Investigate & Apply: Modelling Daylight HoursInvestigate & Apply: Modelling Daylight Hours

Web ConnectionTo find sunrise and sunset data, go towww.mcgrawhill.ca/links/CAF12and follow the links.

Application

Communication


Knowledge/Understanding

534 MHR Chapter 8

Review of Key ConceptsReview of Key Concepts

8.1 Addition and Subtraction FormulasRefer to the Key Concepts on page 496.

1. Express as a single trigonometric function.Then, evaluate.a) sin 64 cos 4 � cos 64 sin 4b) sin 32 cos 13 � cos 32 sin 13c) cos 45 cos 15 � sin 45 sin 15d) cos 45 cos 15 � sin 45 sin 15

e)

8.2 Double-Angle FormulasRefer to the Key Concepts on page 502.2. Express as a single sine or cosine function.a) 50 sin x cos x b) 15 sin 3x cos 3x

c) 2 cos2 (3� � 2) � 1 d) 8 sin cos

e) cos2 2x � sin2 2x f ) 2cos210� � 1

g) 1 � 2 sin2 h) sin 7x cos 7x

3. Solve for x where x � [0, 2�].a) sin 3 cos x � cos 3 sin x � 0.5b) cos x cos � � sin x sin � � 0c) cos 2x cos x � sin 2x sin x � 0d) sin 2x cos x � cos 2x sin x � 1e) sin x cos x � �1 � cos x sin x

f )

g) 3 sin 2x cos x � 3 sin2x � 0h) sin 2x � sin x � 0i) sin x � cos x � 0j) 3 cos 2x � 2 � 0k) sin 2x � tan x � 0l) sin 2x � cos 2x � 0m) 2 sin 2x � cos xn) 3 sin x � 2 � cos 2x

4. Suppose that x � , y � ,

sin x � , and tan y � � . Determine the

value of each of the following.a) sin 2(x � y) b) cos 2(x � y)c) tan 2(x � y) d) sin 2x � cos 2x

5. Marion estimates that the angle ofelevation to the edge of an 11 m cliff is half the

angle of elevation to the top of the 13 m treethat is on the edge of the cliff. How faraway from the base of the cliff is Marionstanding?

6. Determine the value of sin 2(x � y), if

sin x � , x � , cos y � � , and

y � .

8.3 Limits of Trigonometric Functions

Refer to the Key Concepts on page 508.


a) b)

c) d)

e)

8.4 Derivatives of the Sine, Cosine, andTangent Functions

Refer to the Key Concepts on page 517.

8. Determine the derivative of y with respect tox for each function.a) y � sin(3x2

� 5) b) y � 4x3sin2(6x2� 2)

c) y � x2cos(4x2� 7) d) y � sin2(cos3x � tan x)

e) y � sin3(cosx3) f ) y � cos(ex)g) y � cos(ln(tan x)) h) y � ln(sin (ex))

i) y � j) y � sin2x � cos2 x

9. Determine the points of inflection and thelocal maximum and minimum values of the func-tion y � 2 sin x � sin2 x in the domain [0, 2�].

10. Determine the equation of the line that istangent to the graph of y � 2 sin x � sin2x at

the point .�6

54

,

(sin )xx

lim tanx

x xx→

−0 2

lim tanx

xx→ 0

2

22lim sin cosx

x xx→ 0

limsin

x

x

x→ 0

12lim sin

x

xx→ 0

34

��

2,

45

02

, �

513

11 m

13 m

x

x

34

35

��

2,

0

2, �

tan tan

tan tan

x

x

−

+=

�

�2

12

1

25�

x2

x2

tan tantan tan35 10

1 35 10° + °

− ° °

Review of Key Concepts MHR 535

8.5 Modelling With TrigonometricFunctionsRefer to the Key Concepts on page 529.

11. A pulsar, thought to be a rapidly spinningneutron star, emits pulses of “light,” in somewhatthe same way that a lighthouse emits light. (The“light” emitted could be some combination oflight, radio waves, X-rays, etc.) One of the mostfamous pulsars is the power source for the Crabnebula. This pulsar has a rotational period ofabout 0.033 s.a) Assuming that the rate of rotation of theCrab pulsar is constant, use a sinusoidal functionto model the pulses. Assume two pulses perrotation.b) The actual rate of rotation of the Crab pulsaris decreasing so that it will stop in about 2500years. Modify the model in part a) to take thisnew information into account.c) The Crab pulsar is about 6 � 1016 km fromEarth. At what rate does the light from thepulsar “sweep” across the face of Earth?

12. A trough is made from a sheet of metal 3 mwide by bending up one third of the sheet oneach side through an angle �. What value of �will maximize the capacity of the trough?

13. Two sides of a triangle have lengths 15 mand 20 m. The angle between them is increasing

at a rate of rad/s. At what rate is the area

changing when the angle between the two given sides is ?

14. The period of a pendulum is nearlyindependent of its amplitude and its mass,

although it does depend on its length according

to the formula , where L is the

length of the pendulum, in metres, and g � 9.8 m/s2.a) For a certain grandfather clock, theperiod of the pendulum is 1 s. Determine thelength of a pendulum that has this period.b) Is the length found in part a) typical ofgrandfather clocks? If there is a difference, whatdo you think is the reason for the difference?c) Construct a sinusoidal function that models apendulum that has the period specified in part a).d) The maximum amplitude of the pendulumoccurs when the angle between the pendulumand a vertical line is 30. Use this information todetermine one of the constants in the formula ofpart c).e) Determine the maximum angular speed ofthe pendulum and when it occurs.f ) Determine the maximum angularacceleration of the pendulum and when it occurs.

15. The surface area of one cell of a bee’shoneycomb is given by

where � is the angle of inclination at the base ofthe cell, a is the length of a hexagonal side, andb is the average depth of the cell. For a givena and b, determine the angle � that minimizesthe surface area. Note that honey bees actuallyconstruct their cells at this angle.

S ab a( )cos

sin�

�

�= + −

6 32

32

T Lg

� 2�

�3

�90

�

1 m1 m

1 m�

Web ConnectionTo learn more about pulsars, go towww.mcgrawhill.ca/links/CAF12and follow the links.

536 MHR Chapter 8

Chapter Test

3, 5, 73, 73, 4, 6, 7AllQuestions

ApplicationCommunicationThinking/Inquiry/Problem SolvingKnowledge/UnderstandingCategory

Achievement Chart


a) b)

2. Determine and in each case.

a) y � 2 sin 3x � 3 cos 4x b)

3. The blood pressure of a person at rest isdescribed by the formula P � 100 � 20 sin 6t,where P is the pressure, in millimetres ofmercury (mmHg), and t is the time, in seconds.a) Determine the rate at which the heart beats,in beats per minute, assuming that one completecycle of the pressure function corresponds toone heartbeat.b) Determine the maximum and minimumvalues of the pressure. Are these healthy valuesfor blood pressure?c) How is the formula for blood pressure likelyto change during times of physical exertion orwhile taking a math test?d) How is the formula for blood pressure likelyto change during times of relaxation or sleep?

4. Determine the values of n for which the nthderivative of the function f (x) � sin x � cos x isequal to f (x).

5. The cross section of a trough is an invertedisosceles triangle. Determine the vertex angle forwhich the capacity of the trough is a maximum ifthe equal side lengths of the triangle are constant.

6. A circle of radius 1 is centred at theorigin O. Lines drawn from a point (p, 0)are tangent to the circle at points A and B.

a) Determine the value of p for which the areaof �OAB is a maximum.b) Determine the angle between OA and thex-axis for which the area of �OAB is amaximum.c) Determine the maximum area of �OAB.

B

–1

1 A

2 xO

–1

1

y

p

�

yx

x= +

sintan1

d ydx

2

2dydx

lim tanx

xx→ 0

20lim sinsinx

xx→ 0

2

Application

Communication


Knowledge/Understanding 7. The position function of a particle that moves along the x-axis isx � 2�t � sin 2�t.a) Determine a formula for the velocity of the particle.b) Determine a formula for the acceleration of the particle.c) When is the particle momentarily at rest?d) When is the particle moving toward the left? toward the right?e) When is the velocity of the particle increasing? decreasing? f ) When is the particle speeding up? slowing down?g) Sketch the graphs of the position function, the velocity function,and the acceleration function for the particle.


Challenge Problems MHR 537

1. a) Express 4 sin x � 3 cos x in the form A sin (x � �).b) Show that it is always possible to write a sin x � b cos x in the form A sin (x � �) by determiningexpressions for A and � in terms of a and b.

2. a) Express 4 sin x � 3 cos x in the form A cos (x � �).b) Show that it is always possible to write a sin x � b cos x in the form A cos (x � �) by determiningexpressions for A and � in terms of a and b.

3. Prove that is an identity.

4. Find the acute angle of intersection between the lines y � 2x � 1 and y � �4x � 5.

5. Find the acute angle of intersection between the two curves y � x3 and y � x2� 4.

6. If A, B, and C are angles in a triangle, by expanding tan (A � B � C), show thattan A tan B tan C � tan A � tan B � tan C.

7. Determine .

8. Determine .

9. Show that the curves y � e�x and y � e�x cos x are tangent to each other at each intersection point.Sketch the two curves.

10. The total electromotive force E in a circuit is related to the current I by , where

I and r are constants of the circuit. Given that are constants,determine a formula for E.

I I wt b I w b� �0 0cos( ), , , and and

I dIdt

rI E� �

limsin

sinx

x

x

� �

2

1

lim tan sinx

x xx→

+ − +0 3

1 1

sin sin sincos cos cos

tanx x xx x x

x+ ++ + =2 3

2 32

Challenge Problems

extension: trigonometric functions and their...

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