- The document discusses the unit circle and relates angles to their sine and cosine values. - It defines the unit circle as having a radius of 1 and identifies the sine and cosine of angles in all four quadrants. - Key properties discussed are that the sine and cosine of angles in quadrants 2, 3 and 4 can be related to the sine and cosine of the corresponding reference angle in quadrant 1.

1. Unit Circle
• Recognise the properties of the unit circle.
• Identify the sine and cosine of an angle in the unit circle.
• Define the sine and cosine for obtuse and reflex angles.
• Relate the sine and cosine of angles in quadrants 2,3 and 4 with the sine
and cosine of a reference angle in the first quadrant.
By the end of the lesson you will be able to:

2. Quadrant IQuadrant II
Quadrant III Quadrant IV
x
y

3. Quadrant IQuadrant II
Quadrant III Quadrant IV
x
y
x positive
y positive
x negative
y positive
x positivex negative
y negative
y negative
0o
0 rad
90o
π/2 rad
180o
π rad
270o
3π/2 rad

4.
A
.P
θ
Angle measurement
P lies anywhere on the unit circle and A = (1,0).
Angle θ will be the angle measured from

5. Represent the following angles on the unit circle.
θ = 1350
θ = 2400
θ = 3000
θ = ­ 600
Unit circle 2.ggb

6. Mark on the unit circle the points corresponding to
the angles:
A: 135o
D: 450o
C: ­120o
B: 270o

7.
Complete the table:
angle in degrees
30o
225o
390o
angle in radians quadrant

8. Mark on the unit circle the points corresponding to
the angles:
A:
D: C:
B:

9. Draw the angles:

11. Book page 182, Ex. 8A
to revise radians, angles
in the unit circle

12. The Incredible Unit Circle
r=1

13. • P ( x , y )
What is the relationship between the coordinates of a point P on the
unit circle?
State the coordinates of any point on the circumference.
r =1

14. Does the point belong to the unit circle?
x2
+ y2
= 1

15. Does the point belong to the unit circle?
x2
+ y2
= 1
.P
y
Is there any other point on the unit circle where the x coordinate is ?

16. What is the relationship between the coordinates of a point P on the
unit circle?
x
• P ( x , y )
y
1
Therefore , the equation of the unit circle
is: x2
+ y2
= 1
Unit circle 1.ggb
What will be the equation of a circle of centre in the origin and
radius 2?

17. Is there any point P(x,y)on the unit circle where x = √2?
What are the possible values of x ?
What are the possible values of y ?
x2
+ y2
= 1
http://science.kennesaw.edu/~plaval/applets/TrigDef.html

18. y = sin θ
P(x,y)
1
θ
x =cos θ
Express x and y in terms of angle θ

19. When P is on any quadrant:
.P(x,y)
x =cos θ
y = sin θ θ

20. lete the table:
P
(1,0)
(0,­1)
θ
cos θ
-1
sin θ
Unit circle all quadrants.ggb
Complete the table:

21. y = sin θ
P(x,y)
1
x =cos θ
θ
Some important conclusions:
­1 ≤ sin θ ≤ 1
­1 ≤ cos θ ≤ 1
sin2
θ + cos2
θ = 1

22. Signs of sine and cosine
drag the orange line around the circle

23. Use the half unit circle to :
sin 40o
cos 120o
sin 150o cos 10o
2)Find two different angles with the same sine.
1) Estimate the following
3) Find an angle with a cosine of 0.5.
4) Mark an angle whose sine is 0.2. Is the answer unique?

24. Use the unit circle to find : a) sin 50o b) cos 50o

25. sin 50o
≅0.77
cos 50o
≅0.64

26. a) sin 230o b) cos 230o
Given that sin 50° ≈0.77 and cos 50°≈0.64, use the unit circle to
find:

27. Use the unit circle to find : a) sin 230o b) cos 230o
cos 230o
≅- 0.64
sin 230o
≅ -0.77
We can relate any angle in the third quadrant with one in the first
quadrant. Thus
230° =180o
+50o

28. cos α = ­ cos ( α +π)
sin α = ­ sin ( α +π)
Conclusions:
απ+α
If α is in the first quadrant, π+α represents an angle of the third quadrant.

29. If α is in the first quadrant then (π + α) represents
an angle in the third quadrant.
sin ( α +π) = ­ sin α
cos ( α +π) = ­ cos α
Example:
=
sin 225o
=
cos 210o
=

30. Knowing that and use the unit circle to find :
a) sin b) cos 315o
315o

31. Use the unit circle to find : a) sin b) cos 315o
315o

32. In general :
Conclusions:
cos α = cos ( 2π­α) or cos α = cos (­α)
sin α = ­ sin ( 2π­α) or sin α = ­ sin (­α)
α
2π-α
If α is in the first quadrant, 2π­α represents an angle of the fourth quadrant.

33. Conclusions:
cos ( 2π­α) = cos α or cos (­α) = cos α
If α is in the first quadrant, 2π ­ α represents an
angle of the fourth quadrant.
sin ( 2π­α) = ­sin α or sin (­α)= ­ sin α
Example:
sin 300o
=

34. α is an angle in the first quadrant and β is in the second quadrant such that
sin α = sin β
cos α= ­cos β
What is the relationship between α and β?

35. α is an angle in the first quadrant and β is in the second quadrant such that
sin α = sin β
cos α= ­cos β
What is the relationship between α and β?
sin ( π­ α) = sin α
cos ( π­ α) = ­ cos α
α+β=π ⇒ β= π-α
Conclusions:
If α is in the first quadrant, π­α represents an angle of the second quadrant.
α
β

36. Conclusions:
If α is in the first quadrant, π­α represents an angle of
the second quadrant.
sin ( π­ α) = sin α
cos ( π­ α) = ­ cos α
Example:
sin 120o
=

37. Mark all the angles α, 0 ≤ α ≤ 2π, such that cos α = -0.4

38. a) Mark all the angles α, 0≤ α ≤2π, such that cos α = -0.4
b) Estimate the values of α
c) Use the unit circle to estimate sin α in each case.
e) Calculate the values of α, correct to 3 s.f.
d) Use Pythagoras' Identity to calculate the exact value of
sin α in each case.

39. Use the unit circle to find the values of:

40. Use the unit circle to find the values of:

41. Attachments
Unit circle coordinates.exe
tan curve.exe
2010­03­14_1831.swf
Unit circle.ggb
Unit circle all quadrants.ggb
Unit circle 1.ggb
Unit circle class worksheet 2012.docx
Unit circle 2.ggb