trigonometry

Introduction• In this chapter you will learn about secant,

cosecant and cotangent, based on cosine, sine and tan

• We will also look at the inverse functions of sine, cosine and tan, known as arcsin, arccos and arctan

• We will build on the Trigonometric Equation solving from C2

TrigonometryYou need to know the

functions secantθ, cosecantθ and cotangentθ

6A

1seccos

1cosecsin

1cottan

All 3 are undefined if cosθ, sinθ or

tanθ = 0

1 1xx

1 1coscos

You should remember the index law:

It is NOT written like this in Trigonometry

1cos sec

sintancos

coscotsin

so

Something which will be VERY useful later in the chapter…



6A

1seccos

1cosecsin

1cottan

Example QuestionsWill cosec200 be positive or negative?1cosec200

sin 200

90 180

270 360y = Sinθ

As sin200 is negative, cosec200 will be as well!



6A

1seccos

1cosecsin

1cottan

Example QuestionsFind the value

of:

to 2dpsec 280

1sec 280cos 280

sec 280 5.76

Just use your calculator!



6A

1seccos

1cosecsin

1cottan

Example QuestionsFind the value

of:

to 2dpcot115

1cot115tan115

cot115 0.47

Just use your calculator!

-60



6A

1seccos

1cosecsin

1cottan

Example QuestionsWork out the exact value of:

(you may need to use surds…)

sec 210

1sec 210cos 210

y = Cosθ90 18

0270 360

210

30-60

By symmetry, we will get the same value for cos210 at cos30 (but with the reversed sign)

1sec 210cos30

1sec 2103

2

2sec 2103

2 3or 3

Cos30 = √3/2

Flip the denominator



6A

1seccos

1cosecsin

1cottan

Example QuestionsWork out the exact value of:

(you may need to use surds…)

3cosec4

3 1cosec34 sin4

3π/4

Sin(3π/4) = Sin(π/4)

Sin(π/4) = Sin45 1/√2

π/2 π 3π/2 2πy = Sinθ

π/4

3 1cosec4 sin

4

3 1cosec 142

3cosec 24

Flip the denominator


graphs of secθ, cosecθ and cotθ

6B

1seccos

1cosecsin

1cottan

90 180

270 360y = Sinθ

1

0-1

y = Cosecθ

1cosecsin

At 90°, Sinθ = 1 Cosecθ = 1

At 180°, Sinθ = 0 Cosecθ =

undefined We get an asymptote wherever

Sinθ = 0



6B

1seccos

1cosecsin

1cottan

90 180

270 360

y = Cosθ

1

0-1

y = Secθ

1seccos

At 0°, Cosθ = 1 Secθ = 1

At 90°, Cosθ = 0 Secθ =

undefined We get asymptotes wherever Cosθ = 0



6B

1seccos

1cosecsin

1cottan

1cottan

90 180

270 360

y = Tanθ

y = Cotθ

At 45°, tanθ = 1 Cotθ =

1 At 90°, tanθ = undefined Cotθ =

0

At 180°, tanθ = 0 Cotθ =

undefined



6B

1seccos

1cosecsin

1cottan

90 180

270 360y = Sinθ

10

-1

90 180

270 360

1

0-1

y = Cosecθ

Maxima/Minima at (90,1) and (270,-1)

(and every 180 from then)

Asymptotes at 0, 180, 360

(and every 180° from then)



6B

1seccos

1cosecsin

1cottan

Maxima/Minima at (0,1) (180,-1) and (360,1)(and every 180 from

then)

Asymptotes at 90 and 270


90 180

270 360

y = Cosθ

1

0-1

90 180

270 360

1

0-1

y = Secθ



6B

1seccos

1cosecsin

1cottan

Asymptotes at 0, 180 and 360


90 180

270 360

y = Tanθ

90 180

270 360

y = Cotθ

-1



Sketch, in the interval 0 ≤ θ ≤ 360, the graph of:

6B

1seccos

1cosecsin

1cottan

1 sec 2y

90 180

270 360

1

0-1

y = Secθ

secy

sec 2y

1 sec 2y

Horizontal stretch, scale factor 1/2

Vertical translation, 1 unit

up

90 180

270 360

1

0

y = Sec2θ

y = 1 + Sec2θ

2

TrigonometryYou need to be able to simplify expressions, prove identities and

solve equations involving secθ, cosecθ

and cotθ

This is similar to the work covered in C2, but there are

now more possibilities

As in C2, you must practice as much as possible in order to get

a ‘feel’ for what to do and when…

6C

1seccos

1cosecsin

1cottan

Example QuestionsSimplify

…sin cot sec

sin cos sin

1

cos

sin cos sin cos

1

Remember how we can rewrite cotθ from earlier?

sintancos

coscotsin

Group up as a single

fraction

Numerator and

denominator are equal



and cotθ

This is similar to the work covered in C2, but there are

now more possibilities

As in C2, you must practice as much as possible in order to get

a ‘feel’ for what to do and when…

6C

1seccos

1cosecsin

1cottan

Example QuestionsSimplify

… sin cos sec cosec

Rewrite the part in brackets

sintancos

coscotsin

1 1sin coscos sin

sin cossin cossin cos sin cos

sin cossin cossin cos

sin cos sin cossin cos

sin cos

Multiply each fraction by the opposite’s

denominator

Group up since the denominators are now

the same

Multiply the part on top by the part

outside the bracket

Cancel the common factor to

the top and bottom

Trigonometry

6C

1seccos

1cosecsin

1cottan

Show that:

32 2

cot cosec cossec cosec

Rewrite both

sintancos

coscotsin

2 2

cot cosecsec cosec

Left side

cot cosec Numerat

orDenominat

or

cossin

1 sin

2

cossin

Group up

2 2sec cosec

2 2

1 1cos sin

2 2

2 2 2 2

sin coscos sin cos sin

2 2

2 2

sin coscos sin

Rewrite both

Group up

Multiply by the

opposite’s denominat

or

2 2

cot cosecsec cosec

2 2

1cos sin

From C2 sin2θ+ cos2θ = 1

2

2 2

cossin

1cos sin

2

cossin

2 2

1 cos sin

2 2cos sin 1

2

cossin

Putting them together

Replace numerator

and denominat

or

This is just a division

Change to a multiplicatio

n

3 2

2

cos sinsin

3cos

Group up

Simplify



and cotθ

You can solve equations by rearranging them in terms of

sin, cos or tan, then using their respective graphs

6C

1seccos

1cosecsin

1cottan

Example Question

sintancos

coscotsin

Solve the equation:

sec 2.5 In the range:

0 360

1 2.5cos

sec 2.5

1 cos2.5

cos 0.4

1cos 0.4

90 180

270 360

y = Cosθ

10

-1

113.6

246.4

Rewrite using cos

Rearrange

Work out the fraction

Inverse cos

Work out the first answer. Add 360 if not in the range we

want…Subtract from 360 (to find the equivalent value in the range



and cotθ



6C

1seccos

1cosecsin

1cottan

Example Question

sintancos

coscotsin

Solve the equation:

cot 2 0.6 In the range:

0 360

cot 2 0.6 Rewrite

using tan 1tan 20.6

0 360

0 2 720

Remember to adjust the acceptable

range for 2θ

1 12 tan0.6

Inverse tan

2 59.04

Work out the first value, and others in

the original range (0-360)

y = Tanθ90 18

0270 360

, 239.04You can add 180 to these as the period of tan is

180 2 419.04 , 599.04

29.5, 120, 210, 300 Divide all by 2 (answers to

3sf)



and cotθ



6C

1seccos

1cosecsin

1cottan

Example Question

sintancos

coscotsin

Solve the equation:In the

range:0 360

Rewrite each side

Cross multiply

Divide by Cosθ

Divide by 2

Rewrite the right-hand

side

Trigonometry

6D

1seccos

1cosecsin

1cottan

sintancos

coscotsin

5tan12

A

Given that:

and A is obtuse, find the exact value of secA

Example Question

θ

tan OppAdj

5

12

13

cos AdjHyp

Ignore the negative, and use Pythagoras to work out the missing

side…

12cos13

90 180

270 360

y = Cosθ

10

-1

12cos13

13sec12

Replace A and H from the triangle…

A is obtuse (in the 2nd quadrant)

Cos is negative in this range

Flip the fraction to get Secθ

Trigonometry

6D

1seccos

1cosecsin

1cottan

sintancos

coscotsin

5tan12

A

Given that:

and A is obtuse, find the exact value of

cosecA

Example Question

θ

tan OppAdj

5

12

13

sin OppHyp

Ignore the negative, and use Pythagoras to work out the missing

side…

5sin13

90 180

270 360

10

-1

5sin13

13cosec5

Replace A and H from the triangle…

A is obtuse (in the 2nd quadrant)

Sin is positive in this range

Flip the fraction to get Secθ

y = Sinθ

TrigonometryYou need to know and be able to use

the following identities

You might be asked to show where these come from…

6D

1seccos

1cosecsin

1cottan

sintancos

coscotsin

2 21 tan sec 2 21 cot cosec

2 2sin cos 1

2 2

2 2 2

sin cos 1cos cos cos

2tan 1 2 sec

Divide all by cos2θ

Simplify each part

TrigonometryYou need to know and be able to use

the following identities

You might be asked to show where these come from…

6D

1seccos

1cosecsin

1cottan

sintancos

coscotsin


2 2sin cos 1

2 2

2 2 2

sin cos 1sin sin sin

1 2 cot 2 cosec

Divide all by sin2θ

Simplify each part

Trigonometry

6D

1seccos

1cosecsin

1cottan

sintancos

coscotsin


Prove that:Example Question

24 4

2

1 coscosec cot1 cos

4 4cosec cot

2 2 2 2cosec cot cosec cot

2 2 2 2cosec cot 1 cot cot

2 2cosec cot

2

1sin

2

2

cossin

2

2

1 cossin

2 2sin cos 1

2

2

1 cos1 cos

Factorise into a double bracket

Replace cosec2θ

1The second bracket =

1

Left hand side

Rewrite

Group up into 1

fraction

Rearrange the bottom (as in C2)

Trigonometry

6D

1seccos

1cosecsin

1cottan

sintancos

coscotsin


Prove that:Example Question

2 2 2 2sec cos sin (1 sec )

2 2sin cos 1

2 2sin (1 sec )

2 2 2sin sin sec

2 22

1sin sincos

22

2

sinsincos

2 2sin tan

21 cos 2 sec 1

2 2sec cos

Multiply out the bracket

Right hand side

Replace sec2θ

Rewrite the second term

Replace the fraction

Rewrite both terms based on the inequalities

The 1s cancel out…

This requires a lot of practice and will be slow to begin with. The more questions

you do, the faster you will get!

Trigonometry

6D

1seccos

1cosecsin

1cottan

sintancos

coscotsin


Solve the Equation:

Example Question 24cosec 9 cot

2 2sin cos 1

24cosec 9 cot

24 1 cot 9 cot

24 4cot 9 cot

24cot cot 5 0

(4cot 5)(cot 1) 0

5cot4

cot 1 or

4tan5

tan 1 or

in the interval: 0 360

y = Tanθ90 18

0270 360

38.7, 219 135, 315

Replace cosec2θ

4/5

-1

A general strategy is to replace terms until they are all of the same type (eg

cosθ, cotθ etc…)

Multiply out the bracket

Group terms on the left

sideFactorise

Solve

Invert so we can use the tan graph

Use a calculator for the first answer

Be sure to check for others in the given range

Trigonometry

6E

Undefined√311/√3 or √3/30Tanθ

00.51/√2 or √2/2√3/21Cosθ

1√3/21/√2 or √2/20.50Sinθ

90°60°45°30°0°

Copy and complete, using surds where appropriate…

Trigonometry

6E

Undefined√311/√3 or √3/30Tanθ

00.51/√2 or √2/2√3/21Cosθ

1√3/21/√2 or √2/20.50Sinθ

π/2π/3

π/4π/60

The same values apply in radians as well…

TrigonometryYou need to be able to use the

inverse trigonometric functions, arcsinx, arccosx

and arctanx

These are the inverse functions of sin, cos and tan respectively

However, an inverse function can only be drawn for a one-to-one

function

(when reflected in y = x, a many-to-one function would become

one-to many, hence not a function)

6E

π/2-π/2

-1

1

1

π/2

-π/2

-1

y = sinx

y = arcsinx

y = sinxDomain: -π/2 ≤ x ≤ π/2Range: -1 ≤ sinx ≤

1

y = arcsinxDomain: -1 ≤ x ≤ 1Range: -π/2 ≤ arcsinx ≤

π/2

y = x

Remember that from a function to its inverse, the domain and range

swap round (as do all co-ordinates)



and arctanx



function



6E

π/2-1

1

π

-1y = cosx

y = arccosx

y = cosxDomain: 0 ≤ x ≤ π

Range: -1 ≤ cosx ≤ 1

y = arccosxDomain: -1 ≤ x

≤ 1Range: 0 ≤ arccosx ≤ π

y = x

Remember that from a function to its inverse, the domain and range

swap round (as do all co-ordinates)

π/2

π

1

We can’t use –π/2 ≤ x ≤ π/2 as the domain for cos,

since it is many-to-one…



and arctanx



function



6E

y = tanxDomain: -π/2 < x < π/2 Range: x ε

R

y = arctanxDomain: x ε R

Range: -π/2 < arctanx < π/2

y = tanx

y = arctanx

π/2-π/2

π/2

-π/2

Subtle differences… The domain for tanx cannot equal

π/2 or –π/2

The range can be any real number!

Trigonometry

6E

1

-π/2

-1

y = arcsinx -1

y = arccosx

π/2

π

1

y = arctanx

π/2

-π/2

π/2



and arctanx

Work out, in radians, the value of:

6E

arcsin(0.5)

1sin (0.5)

arcsin(0.5)

6

30

Arctan just means inverse

sin…Remember the

exact values from earlier…



and arctanx


6E

arctan( 3)

1tan ( 3)

arctan( 3)

3

60

Arctan just means inverse

tan…Remember the

exact values from earlier…



and arctanx


6E

2arcsin2

Arcsin just means inverse

sin…

Ignore the negative for now, and remember the

values from earlier…2arcsin2

1 2sin2

1 2sin2

4

1 2sin2

4

45

Sin(-θ) = -Sinθ(or imagine the Sine

graph…)

π/2-π/2

-1

1 y = sinx

You need to be able to use the inverse trigonometric

functions, arcsinx, arccosx and arctanx


√2/2

-√2/2

π/4

-π/4



and arctanx


6E

Arcsin just means inverse

sin…

cos arcsin 1

You need to be able to use the inverse trigonometric

functions, arcsinx, arccosx and arctanx


cos arcsin 1

1cos sin 1

cos2

cos2

0

Think about what value you need

for x to get Sin x = –1

π/2-π/2

-1

1 y = sinx

Cos(-θ) = Cos(θ)

π/2-π/2

-1

1Remember it, or read from the graph…

y = cosx

Summary

6E

• We have learnt about 3 new functions, based on sin, cos and tan

• We have seen some new identities we can use in solving equations and proof

• We have also looked at the inverse functions, arc sin/cos/tanx

trigonometry

Documents

sec trigonometryyou

graphs of sec

undefined cot

undefined trigonometryyou

functions secant

surds y

cot sketch

denominator trigonometryyou