1. Section 1.3
Reference Angles

2. Objectives:
1. To find reference angles.
2. To use reference triangles to determine trigonometric ratios.

3. Definition
A reference angle is the angle formed by the terminal ray of the given angle and the x-axis.

4. By drawing a perpendicular segment from any point on the terminal ray to the x-axis, you will form a reference triangle.

5. EXAMPLE 1 Find sin 210º. x y
 = 210º - 180º
= 30º
210° 

6. EXAMPLE 1 Find sin 210º. 1 2 x y
sin 210º = - 3 -
30º -1 2

7. Practice Question: Find sin 300º (round to the nearest ten thousandth).≈
- 3 2 x y 2 1 3 -
60º

8. Practice Question: Find cot 225º.
= 1 -1 x y -1 2
45º

9. EXAMPLE 2 Find tan 61º40. 40 61º40 = (61 + )º = 61.67º 60
61º40 = ( )º = 61.67º 40 60
tan 61º40 = tan 61.67º ≈ 1.855

10. Practice Question: Find cos 81º15 (round to the nearest ten thousandth).
81º15 = ( )º = 81.25º 15 60
cos 81º15 = cos 81.25º ≈ .1521

11. EXAMPLE 3 Find cos 118º.
cos 118º ≈

12. Practice Question: Find sin 207º (round to the nearest ten thousandth).

13. EXAMPLE 4 Find csc 63º. csc 63º = (sin 63º)-1 ≈ 1.1223
TI-83 Calculator steps (use degree mode):
sin(63)-1[ENTER]

14. Practice Question: Find sec 126º (round to the nearest ten thousandth).
sec 126º = (cos 126º)-1 ≈
TI-83 Calculator steps:
cos(126)-1[ENTER]

15. or  -  =   =   - 180° =  or  -  =  or 2 -  = 
2nd quad
180° -  =  or
 -  = 
1st quad
 =  y x
3rd quad
 - 180° =  or
 -  = 
4th quad
360° -  =  or
2 -  = 

16. Homework: pp

17. ►A. Exercises
Give the measure of the reference angle for each angle.
3. 320°
Since 320° is in the fourth quadrant, subtract 320° from 360° giving you a 40° reference angle.

18. ►A. Exercises
Give the measure of the reference angle for each angle. 7. 4 5
Since a 5/4 angle is in the third quadrant, subtract  from 5/4 giving you a reference angle of /4.

19. ►A. Exercises
Use a calculator to find the following ratios.
9. sin 26°20

20. ►A. Exercises
Use a calculator to find the following ratios.
13. csc 12°18

21. ►A. Exercises
Use a calculator to find the following ratios.
15. sec 2.75

22. ►B. Exercises 17. Third quadrant
List the angles that have special angles as reference angles in the given quadrant. Include quadrantal angles with any quadrant they bound. Give the positive measures less than 360°.
17. Third quadrant

23. ►B. Exercises
Use special angles to find the ratios. Do not use a calculator for these exercises csc 315°

24. 3 5
►B. Exercises
Use special angles to find the ratios. Do not use a calculator for these exercises.
27. cos

25. ►B. Exercises
Find the angle measures for 0°    360° that are associated with the ratios given here.
29. tan  = 2.081

26. ►B. Exercises
Find the angle measures for 0°    360° that are associated with the ratios given here.
31. cot  =

27. ►B. Exercises
Find the angle measures for 0°    360° that are associated with the ratios given here.
33. sin  =

28. ■ Cumulative Review
36. Convert 27° to radians (use ).

29. ■ Cumulative Review
37. Give the radian measure of 27° as a decimal.

30. ■ Cumulative Review 38. tan 
The legs of a right triangle are 2 units and 7 units in length, and  is the smallest angle. Find the following trig ratios.
38. tan 

31. ■ Cumulative Review 39. sin 
The legs of a right triangle are 2 units and 7 units in length, and  is the smallest angle. Find the following trig ratios.
39. sin 

32. ■ Cumulative Review 40. cos 
The legs of a right triangle are 2 units and 7 units in length, and  is the smallest angle. Find the following trig ratios.
40. cos 