Section 1 3 Reference Angles Objectives 1 To

Section 1. 3 Reference Angles

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

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

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

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

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

Practice Question: Find sin 300º (round to the nearest ten thousandth). - 3 sin 300º = 2 ≈ -. 8660 y 1 60º 2 x - 3

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

EXAMPLE 2 Find tan 61º 40. 40 61º 40 = (61 + 60 )º = 61. 67º tan 61º 40 = tan 61. 67º ≈ 1. 855

Practice Question: Find cos 81º 15 (round to the nearest ten thousandth). 15 81º 15 = (81 + 60 )º = 81. 25º cos 81º 15 = cos 81. 25º ≈. 1521

EXAMPLE 3 Find cos 118º ≈ -0. 4695

Practice Question: Find sin 207º (round to the nearest ten thousandth). sin 207º ≈ -0. 4540

EXAMPLE 4 Find csc 63º = (sin 63º)-1 ≈ 1. 1223 TI-83 Calculator steps (use degree mode): sin(63)-1[ENTER]

Practice Question: Find sec 126º (round to the nearest ten thousandth). sec 126º = (cos 126º)-1 ≈ -1. 7013 TI-83 Calculator steps: cos(126)-1[ENTER]

2 nd quad 180° - = or - = 3 rd quad - 180° = or - = 1 st quad = y x 4 th quad 360° - = or 2 - =

Homework: pp. 16 -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.

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

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

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

►A. Exercises Use a calculator to find the following ratios. 15. sec 2. 75

►B. Exercises 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

►B. Exercises Use special angles to find the ratios. Do not use a calculator for these exercises. 19. csc 315°

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

►B. Exercises Find the angle measures for 0° 360° that are associated with the ratios given here. 29. tan = 2. 081

►B. Exercises Find the angle measures for 0° 360° that are associated with the ratios given here. 31. cot = 0. 0992

►B. Exercises Find the angle measures for 0° 360° that are associated with the ratios given here. 33. sin = -0. 5446

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

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

■ Cumulative Review 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

■ Cumulative Review 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

■ Cumulative Review 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