1

Circular Functions• Just a fancy name for Trigonometry

– Thinking of triangles as central angles in a circle

• Have done trigonometry ever since grade 9• This slide show assumes a moderate recollection

of previous grades Trig• It assumes you have completed Unit B Transforms

– Although you will survive without Unit B

• Click the small icon of the projection screen to animate this slide show.

• Enjoy

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Intro: Prior Learning

SO/H CA/H TO/ASO/H CA/H TO/A

Is the Greek letter ‘Theta’. Is the Greek letter ‘Theta’. Tend to represent angles by Tend to represent angles by

Greek lettersGreek letters

Hypotenuse

Hypotenuse

Op

pos

ite

Op

pos

ite

AdjacentAdjacent

Hypotenuse

Oppositesin

Hypotenuse

Adjacentcos

Adjacent

Oppositetan

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Intro: Prior Learning 2Hopefully you are familiar with trigonometric values in a table form:

A triangle with a corner angle whose ‘sine ratio’ is 0.5 for example has an opposite side that is half the length (0.5) of the hypotenuse. Since ‘sine’ is just a ratio of the length of the opposite side divided by the length of hypotenuse side of a right angle triangle. In other words, sine is how many hypotenuses fit into the opposite side.

Here are triangles with a corner angle, A, that has a sine of 0.5:

1010 55

A = sin–1(5/10) = 30°

8844

442220201010

A = sin–1(4/8) = 30°A = sin–1(2/4) = 30°A = sin–1(10/20) = 30°

AA

degrees sin cos tan

0 0 1 0

10 0.174 0.985 0.176

20 0.342 0.940 0.364

30 0.500 0.866 0.577

40 0.643 0.766 0.839

50 0.766 0.643 1.192

60 0.866 0.500 1.732

70 0.940 0.342 2.747

80 0.985 0.174 5.671

90 1.000 0.000 undefined

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Know the Trig ratio – Know the shape 1

It is easy to calculate a trig ratio; just the length one side of a right angle triangle divided by the length of another side. So if you are given the lengths of some sides it is easy to calculate.

But the reverse is true also, if you know a trig ratio of the right angle triangle, you know at least the relative shape of the triangle.

Given sin(A) = 0.5 you know the Opposite side from the angle is half the length of the hypotenuse

A

42

A1.5 Or it could be this similar

triangle (the exact same shape).

5

Know the Trig ratio – Know the shape 2

6

Why do we spend so many grades studying triangles again?

7

Measuring anglesYou are familiar with how to measure angles using a protractor and how to find the measure of an angle when given its trig ratio using the inverse functions: cos-1, sin-1, tan-1 on your calculator (or looking up the angle backwards in a table).

Example: on your calculator cos-1(0.5) = 60° which means the angle that has a cosine of 0.5 is 60 °.

degrees sin cos tan

0 0 1 0

10 0.174 0.985 0.176

20 0.342 0.940 0.364

30 0.500 0.866 0.577

40 0.643 0.766 0.839

50 0.766 0.643 1.192

60 0.866 0.500 1.732

70 0.940 0.342 2.747

80 0.985 0.174 5.671

90 1.000 0.000 undefined

But just like you can measure distances in units of metres, or feet, or miles there is another unit by which to measure angles, and it is not degrees!

9

Radians – The universal measure of angles

Only an earthling would use 360 degrees as the measure of the full angle around a circle. Only earth goes around the sun every 360 days (ish), only earthlings count with 10 fingers and use decimals, etc. The proper mathematical and ‘non- prejudiced’ way to measure angles is to use a measure of a central angle called a ‘radian’.

1 radius1 radius

An arc on the circumference of the circle of length 1 radius

= 1 radian or 1= 1 radian or 1rr or just ‘1’ or just ‘1’

If you move around a circle a length of 1 circle radius, you have moved through an angle of 1 ‘radian’ about the centre.

1 radius1 radius

A radian is about 57.3°, or close to 60°

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Radian Conversion

How to convert between Degrees and How to convert between Degrees and ExactExact Radians Radians:

•There are 100 cm to 1 meter

•There are 12 things to a dozen

•There are 2 radians to 360° of a circle

•180° is the same as r

Conversions:Conversions:

Radians = Degrees * r / 180°

Degrees = Radians * 180 °/ r

618

3

180*3030

2180*9090

6

7

18

21

180*210210

135180

*4

3

4

3

45180

*4

1

4

1

r

2.229180

*44

r

Converting an angle to exact radians means Converting an angle to exact radians means the angle will have the angle will have in it in it

11

Unit Circle and Standard Angles

x

y0,1

–1, 0

0, –1

1,0

Point on a circle

r =1

r

=1

We always measure angles in a counter-clockwise direction from the positive x-axis to the terminal arm

It is often nice to work with a ‘Unit Circle’ with a radius of 1. It just makes calculations easier. Superimposed on a grid the unit circle passes through the points :(1, 0); (0, 1); (-1,0); (0,-1)

12

Unit Circle Chart

See how the common radian measures are converted to degrees on this chart.

60° is /3

The points in brackets are discussed later

13

Co-terminal Angles

Coterminal AnglesCoterminal Angles. Standard angles that share the same terminal arm

225225°°

585585°°

Coterminal Angles (Degrees):Coterminal Angles (Degrees):

225225°°

225° + 360° = 585°225° + 360° = 585°

225° + 360° + 360 ° = 945°225° + 360° + 360 ° = 945°

225225° – 360° = –135°° – 360° = –135°

225225° – 360° – 360° = –495°° – 360° – 360° = –495°

Coterminal angles given by: + n*360+ n*360°° where n is any integer (positive or negative)

––135135°°

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Co-terminal Angles

1313/4/4

––33/4/4

Coterminal Angles (Radians):Coterminal Angles (Radians):

55/4/4

55/4 + 2 /4 + 2 = 5 = 5/4 + 8/4 + 8/4 = 13/4 = 13/4/4

55/4 + 2 /4 + 2 + 2 + 2 = 21 = 21 /4 /4

55/4 /4 –– 2 2 = 5 = 5/4 /4 –– 8 8/4 = /4 = ––33/4/4

55/4 /4 –– 2 2 –– 2 2 = = –11–11/4/4

Coterminal angles given by: + n*2+ n*2 where n is any integer (positive or negative)

5 5 /4 /4

Every angle in standard position has an infinite number of co-terminal angles; just depends on how many more times you want to wind or un-wind it!

4

5

4

13

4

21

4

29

4

37

4

3

4

114

19

15

Reference AnglesReference Angle [Reference Angle [RefRef]] : The positive acute (ie: less than 90) angle formed by a terminal arm and the nearest (positive or negative) x-axis. The angle of a simple triangle that the classical Greeks would have discussed.

nearest x-axisrefref

140140°°4040°°

Standard Standard AngleAngle

nearest x-axis

refref

230230°°

5050°°

Standard Standard AngleAngle

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SineSine

Hyp

oten

use

Op

posi

te

Adjacent x

y

0,1

–1, 0

0, –1

1,0

Classical Greek Sine Definition:Classical Greek Sine Definition:

Hypotenuse

Oppositesin

Modern Circular Definition:Modern Circular Definition:

r

ysin

yyrr

The classical and modern definitions of sine agree;

except we now call the hypotenuse a radius, r

and we call the opposite side the ‘y-coordinate’ of a point on the circle

Point on a circle P(x,y)

If we pretend the circle has a radius of one then the sine is just the percentage of how high above or belowhigh above or below the ‘hub’ (ie: centre or origin) of the circle we are.

17

CosineCosine

Hyp

oten

use

Op

posi

te

Adjacent x

y

0,1

–1,0

0,–1

1,0

Classical Greek Cosine Definition:Classical Greek Cosine Definition:

Hypotenuse

Adjacentcos

Modern Circular Definition:Modern Circular Definition:

r

xcos

xx

rr

The classical and modern definitions of cosine agree;

except we call the hypotenuse a radius, r, now

and we call the adjacent side the ‘x-coordinate’ of a point on the circle

Point on a circle (x,y)

If we pretend the circle has a radius of one then the cosine is just the percentage of how right right or leftor left of the centre of the circle we are. The point P(x, y) is therefore

(cos(cos, sin , sin ) ) on the unit circle

coscos, sin , sin

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Special Angles - Quadrantal Angles

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Special Angles 45°– 45° TriangleWe know the value of the sine and cosine of a

45° – 45 ° – 90 ° triangle angles from Geometry!

4545°°

4545°°

11

11

2xx

2

2

112

222

x

x

x

707.0

2

2

2

1)4/sin(45sin

Hyp

Opp

707.0

2

2

2

1)4/cos(45cos

Hyp

Adj

Check with your calculator! sin45° is approx 0.707 and so is

2

2

valueexacttheis2

2

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Special Angles 30°– 60° TriangleWe know the exact value of the sine and cosine of the

30°– 60 – 90° triangle from geometry

3

3

14

12

2

2

222

x

x

x

x

5.02

1)6/sin(30sin

Hyp

Opp2

2

2

1

xx

6060°°

6060°°

3030°°3030°°

866.02

3)6/cos(30cos

Hyp

Adj3

5.02

1)3/cos(60cos

Hyp

Adj

866.02

3)3/sin(60sin

Hyp

Opp

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Exact Trig Values Pattern 0 30 45 60 90

0 /6 /4 /3 /2

sin 0 1

cos 1 0

2

2

2

1

2

3

2

32

2

2

1

0 30 45 60 90

0 /6 /4 /3 /2

sin

cos 2

2

2

1

2

3

2

32

22

02

4

2

0

2

42

1

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Reading Sine and Cosine from Unit Circle Chart

(cos30, sin30)(cos30, sin30)

(cos45, sin45)

(cos60, sin60)(cos60, sin60)

(cos135, sin135)(cos135, sin135)

(cos90, sin90)(cos90, sin90)

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The Unit Circle Exact ValuesLater you will learn to find exact values of any angle. Not just these special ones.

24

Graph of Sin (converting angular motion to grid coordinates)

y

00 22

Angle Angle [radians] [radians]

11

-1-1

sin sin is just how high above the hub of a ferris wheel you are is just how high above the hub of a ferris wheel you are on a unit circle as function of your angleon a unit circle as function of your angle

Sin(Sin())

‘‘Wavelength’ or ‘Period’ of one cycleWavelength’ or ‘Period’ of one cycle

DomainDomain: All ; ie: -- < < < < ; you can go around front wards or backwards as ; you can go around front wards or backwards as many times as you wantmany times as you want

Range:Range: -1 -1 y y 1, you will never go above 1 or below -1 1, you will never go above 1 or below -1

25

Graph of Cosine

-4.2-4

-3.8-3.6-3.4-3.2-3

-2.8-2.6-2.4-2.2-2

-1.8-1.6-1.4-1.2-1

-0.8-0.6-0.4-0.20

0.20.40.60.81

1.21.41.61.82

2.22.42.62.83

3.23.43.63.84

4.24.44.64.85

5.2

-2.2 -2 -1.8 -1.7 -1.5 -1.3 -1.2 -1 -0.8 -0.7 -0.5 -0.3 -0.2 -0 0.17

0.33

0.5 0.67

0.83

1 1.17

1.33

1.5 1.67

1.83

2 2.17

2.33

2.5 2.67

2.83

3 3.17

3.33

3.5 3.67

3.83

4 4.17

‘‘Wavelength’ or ‘Period’ of one cycleWavelength’ or ‘Period’ of one cycle

Angle Angle [radians] [radians]

The period or wavelength is 2The period or wavelength is 2 radians radiansDomainDomain: All ; ie: -- < < < < ; you can go around front wards or backwards as ; you can go around front wards or backwards as many times as you wantmany times as you want

Range:Range: -1 -1 y y 1, you will never go above 1 or below -1 1, you will never go above 1 or below -1

Cosine is just the x-coordinate on the unit circle; how far left or right you are of the ‘hub’

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-3

-2

-1

0

1

2

3

4

-1.1667 -1 -0.8333 -0.6667 -0.5 -0.3333 -0.1667 6.2E-10 0.16667 0.33333 0.5 0.66667 0.83333 1 1.16667 1.33333 1.5 1.66667 1.83333 2 2.16667 2.33333 2.5

Graph of Tangent

3 31 133

33

x

ytan

22

Asym

ptote

Asym

ptote =

=

/2/2

Asym

ptote

Asym

ptote =

3 =

3 /2/2

Wavelength Wavelength or periodor period or 180or 180°°

29

Reciprocal Trigonometric Functions

Remember the Transformations unit? ‘y’s less than one got stretched to big ‘y’s, (Eg: ½ becomes 2)

‘y’s more than 1 got compressed to small ‘y’s. (Eg: 4 becomes ¼)

The reciprocal trig functions are the same idea!

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Reciprocal Trig Function - Cosecant

-4.2-4

-3.8-3.6-3.4-3.2-3

-2.8-2.6-2.4-2.2-2

-1.8-1.6-1.4-1.2-1

-0.8-0.6-0.4-0.20

0.20.40.60.81

1.21.41.61.82

2.22.42.62.83

3.23.43.63.84

4.24.44.64.85

5.2

-2.2 -2 -1.8 -1.7 -1.5 -1.3 -1.2 -1 -0.8 -0.7 -0.5 -0.3 -0.2 -0 0.17

0.33

0.5 0.67

0.83

1 1.17

1.33

1.5 1.67

1.83

2 2.17

2.33

2.5 2.67

2.83

3 3.17

3.33

3.5 3.67

3.83

4 4.17

22

33

yy

Sine curveSine curve

11

22

33

44

y=11

1

1y

y=0.5

25.0

1y

y=0.25

425.0

1y

Smaller fractions on the sine curve, turn into larger numbers on the cosecant curve

1/21/222

1/41/444

Plotting csc in red

asym

pto

te

)sin(

1)csc(

31

Solving Trig Equations

To be done in the ‘Trig Identity’ unitYou will then learn to solve equations like:

4sin(2 + /4) = 2

You might have already detected the answer(s)!

= -/24, 23/24, etc