1. Objectives Students will learn how to: Describe angles
Use radian measure
Use degree measure
Use angles to model and solve real-life problems

2. Vocabulary standard position initial side terminal side
angle of rotation
coterminal angle
reference angle

3. Angle = determined by rotating a ray (half-line) about its endpoint.
Initial Side = the starting point of the ray
Terminal Side = the position after rotation
Vertex = the endpoint of the ray
Positive Angles = generated by counterclockwise rotation
Negative Angles = generated by clockwise rotation
Angle
Angle in standard position

4. Angles are labeled with Greek letters such as (alpha),  (beta), and
 (theta).
as well as uppercase letters A, B, and C
Coterminal(opposite)-have the same initial and terminal sides

5. One unit of measurement for angles is degrees, which are based on a fraction of a circle. Another unit is called a radian, which is based on the relationship of the radius and arc length of a central angle in a circle.
Arc length = radius when  = 1 radian

6. These are the common angles/Radian.
Circumference of Circle:

7. A 360° rotation is a complete rotation
A 360° rotation is a complete rotation. A 180° rotation is one-half of a complete rotation.
Remember!
For an angle θ in standard position, the reference angle is the positive acute angle formed by the terminal side of θ and the x-axis.

9. I Do: Drawing Angles in Standard Position
Draw an angle with the given measure in standard position.
A. 320°
B. –110°
C. 990°
Rotate the terminal side 320° counterclockwise.
Rotate the
terminal side –110°
clockwise.
Rotate the
terminal side 990° counterclockwise.

10. Check It Out! You Do!!
Draw an angle with the given measure in standard position.
A. 210°
B. 1020°
C. –300°
Rotate the terminal
side 210° counter-clockwise.
Rotate the terminal
side 1020° counter-clockwise.
Rotate the terminal
side 300° clockwise.

11. I DO: Finding Coterminal Angles
Find the measures of a positive angle and a negative angle that are coterminal with each given angle.
 = 65°
65° + 360° = 425°
Add 360° to find a positive coterminal angle.
65° – 360° = –295°
Subtract 360° to find a negative coterminal angle.
Angles that measure 425° and –295° are coterminal with a 65° angle.

12. You DO: Finding Coterminal Angles
Find the measures of a positive angle and a negative angle that are coterminal with each given angle.
 = 10°
10° + 360° = 370°
Subtract 360° to find a positive coterminal angle.
10° – 360° = –350°
Subtract a multiple of 360° to find a negative coterminal angle.
Angles that measure 370° and –350° are coterminal with a 410° angle.

13. I Do– Finding Coterminal Angles
For the positive angle 13 / 6, subtract 2 to obtain a coterminal angle

14. You DO – Finding Coterminal Angles
cont’d
For the negative angle –2 / 3, add 2 to obtain a coterminal angle

15. I do : Finding Reference Angles
Find the measure of the reference angle for each given angle.
A.  = 135°
B.  = –105°
–105°
The measure of the reference angle is 45°.
The measure of the reference angle is 75°.

16. You Do: Finding Reference Angles
Find the measure of the reference angle for each given angle.
C.  = 325°
D.  = 105°
E.  = –115°
105°
–115°
The measure of the
reference angle is 75°
The measure of the reference angle is 35°.
The measure of the
reference angle is 65°

17. I DO : Converting Degrees to Radians
Convert each measure from degrees to radians.
A. 85° °
π radians
180° 36 =
17 π 36
A. 90° °
π radians
180° 2 = π 2

18. You Do! Example 1
Convert each measure from degrees to radians.
A. –36° °
π radians
180° 5
= - π 5
B. 270° °
π radians
180° 2 = 3π 2

19. I Do: Converting Radians to Degrees
Convert each measure from radians to degrees. 2π 3 A.
2 π
1 3
radians °
Π radians
= 120° π 6 B. π
1 6
radians °
Π radians
= 30°

20. You Do! Example 2
Convert each measure from radians to degrees. 5π 6 A.
5 π
1 6
radians °
Π radians
= 150° 3π 4
B. -
3 π
1 4
radians °
Π radians
= -135° -

22. Example 4: Finding Values of Trigonometric Functions
P (–6, 9) is a point on the terminal side of  in standard position. Find the exact value of the six trigonometric functions for θ.
Step 1 Plot point P, and use it to sketch a right triangle and angle θ in standard position. Find r.

23. Example 4 Continued
Step 2 Find sin θ, cos θ, and tan θ.

24. Example 4 Continued
Step 3 Use reciprocals to find csc θ, sec θ, and cot θ.

25. Because r is a distance, its value is always positive, regardless of the sign of x and y.
Helpful Hint

26. Check It Out! Example 4
P(–3, 6) is a point on the terminal side of θ in standard position. Find the exact value of the six trigonometric functions for θ. y x
P(–3, 6)  θ
Step 1 Plot point P, and use it to sketch a right triangle and angle θ in standard position. Find r.

27. Check It Out! Example 4 Continued
Step 2 Find sin θ, cos θ, and tan θ.

28. Check It Out! Example 4 Continued
Step 3 Use reciprocals to find csc θ, sec θ, and cot θ.