1. – Angles and the Unit Circle

2. Angles and the Unit Circle
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. Do we remember what this is called?
1. 90° 2. 45° 3. 30°
4. 150° ° °

3. Angles and the Unit Circle
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. Do we remember what this is called?
1. 90° 2. 45° 3. 30°
4. 150° ° °
Standard Position
Solutions 1. 2. 3. 4. 5. 6.

4. The Unit Circle
Let’s pick a point on the unit circle. The positive angle always goes counter-clockwise from the x-axis.
The Unit Circle
Radius is always one unit
Center is always at the origin 1 30 -1 1
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. -1

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

6. Angles and the Unit Circle
Find the measure of the angle.
The angle measures 60° more than a right angle of 90°.
Since = 150, the measure of the angle is 150°.
The angle formed by the terminal side of the angle in standard position and the closest x axis is called the reference angle.
Here the reference angle is 30º

7. Angles and the Unit Circle
Sketch each angle in standard position and find the reference angle for each.
a. 48°
b. 310°
c. –170°
Reference is the same
Reference is 50º
Reference is 10º

8. Let’s Try Some
Draw each angle of the unit circle.
45o
-280 o
-560 o

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

10. What are the angle measurements of each of the four angles we just found?
π/2
90° 0° 2π
180°
360° π
270°
3π/2 10

11. The Unit Circle Let’s look at an example
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. 1
In order to determine the sine and cosine we need a right triangle. 30 -1 1 -1

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

13. The Unit Circle 2 60 1 1 30 P
X- coordinate 30 -1 1
Y- coordinate -1

14. The Unit Circle
You can see why the x co-orodinate is cosine and the y co-ordinate is sine when we overlap the two triangles to create similar triangles. 1 2
The smaller triangle has a hypotenuse of 1 unit, the radius of the unit circle which is half our identity triangle. P 60 1 1 30 -1
The X- coordinate is the horizontal distance of the smaller triangle
The Y- coordinate is the vertical distance of the smaller triangle -1

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

16. Angles and the Unit Circle
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.)

17. Angles and the Unit Circle
(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
longer leg = = The longer leg is times the shorter leg. 3 2 1
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°) = – .

18. Let’s Try Some
Draw each Unit Circle. Then find the cosine and sine of each angle.
45o
120o

19. 45° Reference Angles - Coordinates
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.
( , )
3π/4
( , )
135°
45°
π/4
7π/4
5π/4
225°
315°
( , )
( , ) 19

20. Green Triangle
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

21. 60° Reference Angles - Coordinates
Use the side lengths we labeled on the QI triangle to determine coordinates.
2π/3
π/3
60°
120°
( , )
( , )
5π/3
4π/3
( , )
( , )
240°
300° 21

22. Yellow Triangle
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

23. 30° Reference Angles
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.
π/6
150°
30°
5π/6
11π/6
330°
210°
7π/6 23

24. 30° Reference Angles - Coordinates
Use the side lengths we labeled on the QI triangle to determine coordinates.
( , )
( , )
150°
30°
π/6
5π/6
7π/6
11π/6
330°
210°
( , )
( , ) 24

25. Final Product 25

26. The Unit Circle 26