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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
1
Intro: Prior Learning
 Is the Greek letter ‘Theta’.
Tend to represent angles by
Greek letters
Opposite
SO/H CA/H TO/A

Adjacent
Opposite
sin  
Hypotenuse
Adjacent
cos  
Hypotenuse
Opposite
tan  
Adjacent
2
Intro: Prior Learning 2
Hopefully you are familiar with trigonometric values in a table form:
degrees
0
10
20
30
40
50
60
70
80
90
sin
0
0.174
0.342
0.500
0.643
0.766
0.866
0.940
0.985
1.000
cos
1
0.985
0.940
0.866
0.766
0.643
0.500
0.342
0.174
0.000
tan
0
0.176
0.364
0.577
0.839
1.192
1.732
2.747
5.671
undefined
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:
4
820
10
4210
5
A = sin–1(10/20)
(5/10)==30°
(4/8)
(2/4)
=30°
30°
A
3
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
4
2
A
.5
1
A
Or it could be this similar
triangle (the exact same
shape).
4
Know the Trig ratio – Know the shape 2
5
Why do we spend so many
grades studying triangles again?
6
Measuring angles
You 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 °.
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!
degrees
0
10
20
30
40
50
60
70
80
90
sin
0
0.174
0.342
0.500
0.643
0.766
0.866
0.940
0.985
1.000
cos
1
0.985
0.940
0.866
0.766
0.643
0.500
0.342
0.174
0.000
tan
0
0.176
0.364
0.577
0.839
1.192
1.732
2.747
5.671
undefined
7
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’.

A radian is about
57.3°, or close to
60°
1 radius
An arc on the
circumference of the
circle of length 1 radius
 = 1 radian or 1r 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.
9
Radian Conversion
How to convert between Degrees and Exact Radians:
•There are 100 cm to 1 meter
3
3
180
  *
 135
4
4

•There are 12 things to a dozen
•There are 2 radians to 360° of a circle
•180° is the same as r
3 
30  30 *



180
18 6

21 7


210  210 *



180
18
6

Conversions:
Radians = Degrees * r / 180°
Degrees = Radians * 180 °/ r
90  90 *

1 r 1
180
  *
 45
4
4



180



2

4r  4 *

180

 229.2
Converting an angle to exact radians means
the angle will have  in it
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Unit Circle and Standard Angles
y
0,1

–1, 0
Point on a
circle
1,0
We always measure
angles in a counterclockwise direction
from the positive xaxis to the terminal
arm
x
0, –1
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)
11
Unit Circle Chart
60° is /3
See how the
common radian
measures are
converted to
degrees on this
chart.
The points in
brackets are
discussed later
12
Co-terminal Angles
Coterminal Angles. Standard angles that share the same
terminal arm
Coterminal Angles (Degrees):
585°

225°
225° + 360° = 585°
225°
–135°
225° + 360° + 360 ° = 945°
225° – 360° = –135°
225° – 360° – 360° = –495°
Coterminal angles given by:
 + n*360° where n is any integer
(positive or negative)
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Co-terminal Angles
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!
Coterminal Angles (Radians):
5
13
4
5/4
13/4
4
21
5/4 + 2  = 5/4 + 8/4 = 13/4
4
 5 /4
5/4 + 2  + 2  = 21 /4
29
–3/4
5/4 – 2  = 5/4 – 8/4 = –3/4
4
37
5/4 – 2  – 2  = –11/4
4
Coterminal angles given by:
 3
 19

11

 + n*2 where n is any integer
4
4
(positive or negative)
4
14
Reference Angles
Reference Angle [Ref] : 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.
Standard
Angle
40°
ref
nearest x-axis
Standard
Angle

140°
nearest x-axis

230°
50° ref
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Sine
sin  
y
Opposite
Hypotenuse
Point on a
circle P(x,y)The
0,1
y

–1, 0
Adjacent
Modern Circular Definition:
y
sin  
r
0, –1
If we pretend the circle has a radius of one then the sine is
just the percentage of how high above or below the ‘hub’
(ie: centre or origin) of the circle we are.
Opposite
Classical Greek Sine Definition:
classical and
modern
definitions of
sine agree;
1,0
x
except we now
call the
hypotenuse a
radius, r
and we call the
opposite side the
‘y-coordinate’
of a point on the
circle
16
Cosine
cos  
y
Point on a
cos,
 The classical and
circlesin
(x,y)
0,1
Adjacent
Hypotenuse

–1,0
x
Adjacent
Modern Circular Definition:
x
cos  
r
0,–1
If we pretend the circle has a radius of one then
the cosine is just the percentage of how right
or left of the centre of the circle we are.
Opposite
Classical Greek Cosine Definition:
modern
definitions of
cosine agree;
1,0
x
except we call
the hypotenuse a
radius, r, now
and we call the
adjacent side the
‘x-coordinate’ of
a point on the
circle
The point P(x, y) is therefore
(cos, sin ) on the unit circle 17
Special Angles - Quadrantal Angles
18
Special Angles 45°– 45° Triangle
We know the value of the sine and cosine of a
45° – 45 ° – 90 ° triangle angles from Geometry!
45°
2
1
45°
1
x 2  12  12
x2  2
x  2
sin 45  sin(  / 4) 
Opp
1
2


Hyp
2
2
 0.707
cos 45  cos( / 4) 
Adj
1
2


Hyp
2
2
 0.707
2
is the exact value
2
Check with your calculator!
sin45° is approx 0.707 and so is 2
2
19
Special Angles 30°– 60° Triangle
We know the exact value of the sine and cosine of the
30°– 60 – 90° triangle from geometry
60°
Opp 1
sin 30  sin(  / 6) 
  0.5
Hyp 2

30°
30°
3
x
Adj
3
cos 30  cos( / 6) 

 0.866
Hyp
2

cos 60  cos( / 3) 
2 2  x 2  12
4  x 1
2
60°
sin 60  sin(  / 3) 
Adj 1
  0.5
Hyp 2
Opp
3

 0.866
Hyp
2
x 3
2
x  3
20
Exact Trig Values Pattern

0
30
45
60
90

0
/6
/4
/3
/2
sin
0
1
2
2
2
3
2
1
cos 
1
3
2
2
2
1
2
0

0
30
45
60
90

0
/6
/4
/3
/2
3
2
1
2
4
2
0
2
sin
cos 
0
2
4
2
1
2
3
2
2
2
2
2
21
Reading Sine and Cosine from Unit Circle Chart
(cos90, sin90)
(cos60, sin60)
(cos135, sin135)
(cos45, sin45)
(cos30, sin30)
22
The Unit Circle Exact Values
Later you will learn
to find exact values of
any angle. Not just
these special ones.
23
Graph of Sin
y
1. 2
(converting angular motion to grid coordinates)
Sin()
1. 1
1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
‘Wavelength’ or ‘Period’ of one cycle
0
-1
- 0.
- 0.
- 0.
- 0.
- 0.
- 0.
- 0.
- 0.
- 0.
- 0.
9
8
8
7
6
5
4
3
3
2
--00.. 1 - 0
1
- 0.2
0.0
0
8
0.1 0.2
7
5
0.3
0.4
3
2
0.5

0.5
0.6
0.7
0.8
0.9
8
7
5
3
2
- 0.3
- 0.4
1
1. 0
1. 1
1. 2
1. 3
1. 4
8
7
5
3
2
1. 5
1. 5
1. 6
1. 7
1. 8
1. 9
8
7
5
3
2
2
2
- 0.5
- 0.6
- 0.7
- 0.8
-1
- 0.9
-1
- 1. 1
- 1. 2
Angle  [radians]
sin  is just how high above the hub of a ferris wheel you are
on a unit circle as function of your angle
Domain: All ; ie: - <  < ; you can go around front wards or backwards as
many times as you want
Range: -1  y  1, you will never go above 1 or below -1
24
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
0.2
0
-0.2 -0 0.1 0.3 0.5 0.6 0.8
1.2 -1 -0.8 -0.7 -0.5 -0.3 -0.2
1
-0.4
7
3
7
3
-0.6
-0.8
-1
-1.2
-1.4
-1.6
-1.8
-2
-2.2
-2.4
-2.6
-2.8
-3
-3.2
Domain: All ; ie: - <  <-3.4
; you can go around front wards or
many times as you want -3.6
-3.8
-4
Range: -1  y  1, you will
never
go above 1 or below -1
-4.2
Graph of Cosine
‘Wavelength’ or ‘Period’ of one cycle

Cosine is just the x-coordinate on the unit circle;
how far left or right you are of the ‘hub’
1.1
7
1.3
3
1.5
1.6
7
1.8
3
Angle  [radians]
The period or wavelength is 2 radians
backwards as
25
Graph of Tangent
4
Wavelength
or period
 or 180°
3
2
3 3
1
-1.1667
-1
-0.8333 -0.6667
1
3
-0.5
0
3 3
1
3
-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
2.5
-1
-2
y
tan  
x
-3
28
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!
29
Reciprocal Trig Function - Cosecant
y
4
3
Plotting csc
in red
1/44
2
1
1/22
Sine curve
1
csc( ) 
sin(  )
1
y
4
0.25
asymptote
5.2
5
4.8
4.6
4.4
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
0.2
0
-0.2 -0 0.1 0.3 0.5 0.6 0.8
-2.2 -2 -1.8 -1.7 -1.5 -1.3 -1.2 -1 -0.8 -0.7 -0.5 -0.3 -0.2
-0.4
7
3
7
3
-0.6
-0.8
-1
-1.2
-1.4
-1.6
-1.8
-2
-2.2
-2.4
-2.6
-2.8
-3
-3.2
-3.4
-3.6
-3.8
-4
-4.2
1
2
0.5
1
y  1
y=1
1
y=0.5
y=0.25
y
1

1.1 1.3 1.5 1.6 1.8
7
3
7
3
2

2.1 2.3 2.5 2.6 2.8
7
3
7
3
2
3
3.1 3.3 3.5 3.6 3.8
7
3
7
3
4
4.1
7
3
Smaller fractions on the
sine curve, turn into
larger numbers on the
cosecant curve
30
Solving Trig Equations
To be done in the ‘Trig Identity’ unit
You will then learn to solve equations like:
4sin(2 + /4) = 2
You might have already detected the answer(s)!
 = -/24, 23/24, etc
31