Unit 9.2

2. For each measure, draw an angle with its vertex at the origin of the
coordinate plane. Use the positive x-axis as one ray of the angle.
1. 90° 2. 45° 3. 30°
4. 150° 5. 135° 6. 120°

3. Solutions
1. 2. 3.
4. 5. 6.
For each measure, draw an angle with its vertex at the origin of the
coordinate plane. Use the positive x-axis as one ray of the angle.
1. 90° 2. 45° 3. 30°
4. 150° 5. 135° 6. 120°

4. 1-1
1
-1
The Unit Circle
-Radius is always one
unit
-Center is always at
the origin
-Points on the unit
circle relate to the
periodic function
30
Let’s pick a point on
the unit circle. The
positive angle
always goes
counter-clockwise
from the x-axis.
The x-coordinate of
this has a value of the
cosine of the angle.
The y-coordinate has a
value of the sine of
the angle.
In order to determine the sine and
cosine we need a right triangle.

5. 1-1
1
-1
The angle can also be
negative. If the angle is
negative, it is drawn
clockwise from the x
axis.
- 45

6. Find the measure of the angle.
Since 90 + 60 = 150, the measure of the angle is 150°.
The angle measures 60° more than a right angle of 90°.

7. Sketch each angle in standard position.
a. 48° b. 310° c. –170°

8. Draw each angle of the unit circle.
a.45o
b.-280 o
c.-560 o

9. Definition: A circle centered at the origin with a
radius of exactly one unit.
|-------1-------|
(0 , 0) (1,0)(-1,0)
(0, 1)
(0, -1)

10. 180°
90°
270°
0°
360° 2π
π/2
π
3π/2
0

11. 1-1
1
-1
Let’s look at an example
30
The x-coordinate of
this has a value of the
cosine of the angle.
The y-coordinate has a
value of the sine of
the angle.
In order to determine the sine and
cosine we need a right triangle.

12. 1-1
1
-1
30
Create a right triangle,
using the following rules:
1.The radius of the circle is
the hypotenuse.
2.One leg of the triangle
MUST be on the x axis.
3.The second leg is parallel
to the y axis.
30
60
1
Remember the ratios of a 30-60-90
triangle-
2

13. 1-1
1
-1
30
30
60
1
2
X- coordinate
Y- coordinate
P

14. Find the cosine and sine of 135°.
Use a 45°-45°-90° triangle to find sin 135°.
From the figure, the x-coordinate of point A
is – , so cos 135° = – , or about –0.71.2
2
2
2
opposite leg = adjacent leg
0.71 Simplify.
= Substitute.
2
2
The coordinates of the point at which the terminal side of a 135° angle intersects
are about (–0.71, 0.71), so cos 13 –0.71 and sin 135° 0.71.

15. Find the exact values of cos (–150°) and sin (–150°).
Step 1: Sketch an angle of –150° in
standard position. Sketch a
unit circle.
x-coordinate = cos (–150°)
y-coordinate = sin (–150°)
Step 2: Sketch a right triangle. Place the
hypotenuse on the terminal side
of the angle. Place one leg on the
x-axis. (The other leg will be
parallel to the y-axis.)

16. (continued)
The triangle contains angles of 30°, 60°, and 90°.
Step 3: Find the length of each side of the triangle.
hypotenuse = 1 The hypotenuse is a radius of the unit circle.
shorter leg = The shorter leg is half the hypotenuse.
1
2
1
2
3
2
longer leg = 3 = The longer leg is 3 times the shorter leg.
3
2
1
2
Since the point lies in Quadrant III, both coordinates are negative. The longer leg
lies along the x-axis, so
cos (–150°) = – , and sin (–150°) = – .

17. Draw each Unit Circle. Then find the cosine and sine of each angle.
a.45o
b.120o

18. Remember that the unit circle is overlayed on a coordinate plane (that’s
how we got the original coordinates for the 90°, 180°, etc.)
Use the side lengths we labeled on the QI triangle to determine
coordinates.
45°135°
315°225°
( , )( , )
( , ) ( , )
2
2
−
2
2
2
2
−
2
2
2
2
−
2
2
−
2
2
2
2
2
2
2
2
π/4
3π/4
5π/4 7π/4

19. Holding the triangle with the single fold down and double fold to the left,
label each side on the triangle.
Unfold the triangle (so it looks like a butterfly) and glue it to the white circle
with the triangle you just labeled in quadrant I, on top of the blue butterfly.

20. Use the side lengths we labeled on the QI triangle to determine
coordinates.
60°120°
300°240°
( , )( , )
( , ) ( , )
2
3
−
2
3
2
1
−
2
3
2
1
−
2
3
−
2
1
2
1
2
1
2
3
π/32π/3
4π/3 5π/3

21. Holding the triangle with the single fold down and double fold to the left,
label each side on the triangle.
Unfold the triangle (so it looks like a butterfly) and glue it to the white circle
with the triangle you just labeled in quadrant I, on top of the green
butterfly.

22. We know that the quadrant one angle formed by the triangle is 30°.
That means each other triangle is showing a reference angle of 30°.
What about in radians?
Label the remaining three angles.
30°150°
330°
210°
π/6
7π/6
5π/6
11π/6

23. Use the side lengths we labeled on the QI triangle to determine
coordinates.
30°150°
330°
210°
( , )( , )
( , ) ( , )
2
1
−
2
1
2
3
−
2
1
2
3
−
2
1
−
2
3
2
3
2
3
2
1
π/6
7π/6
5π/6
11π/6

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