Trigonometric Ratios of Any Angle Chapter 4 Trigonometric Functions 4.2 Trigonometric Ratios of Any Angle MATHPOWERTM 12, WESTERN EDITION 4.2.1

Trigonometric Ratios of Angles in Standard Position When given an angle q in standard position and P(x, y), which is any point on the terminal arm, the three primary ratios are defined as follows: By the Pythagorean Theorem: The three are defined as follows: 4.2.2

Finding Trigonometric Ratios of Angles in Standard Position The point A(-4, 3) lies on the terminal arm of an angle q in standard position. Determine the exact value of the trigonometric ratios for angle q. The reciprocal ratios follow from the primary ratios: 4.2.3

Finding Trigonometric Ratios of Angles in Standard Position . . . . 4.2.4

Reference Angles 300 300 Principal angle _______ 4.2.5

Reference Angles [cont’d] Principal angle _______ Principal angle _______ Principal angle _______ Principal angle _______ 4.2.6

Reference Angles Principal angle _______ Principal angle _______ 4.2.7

Reference Angles Principal angle _______ Principal angle _______ 4.2.8

Reference Angles and the CAST Rule The trigonometric ratios of any angle can be written as the same function of a positive acute angle called the reference angle. When solving for angles greater than 900, the reference angle is used to find the related trigonometric ratio. The is used to determine the sign of the ratio. Find the value of the following trig ratios to 4 decimal places: CAST Rule a) sin 1500 b) sin 2100 Quadrant II Quadrant I Find the measure of q: 0 ≤ q < 3600 a) cos q = -0.6691 The reference angle is . The angle q would be found in Quadrant III Quadrant IV 4.2.9

Exact Values of Special Angles 300, 600, and 900 sin 300 = sin 600 = 300 cos 300 = cos 600 = 600 tan 300 = tan 600 = 450, 450, and 900 sin 450 = 450 cos 450 = 450 tan 450 = 4.2.10

Exact Values of Special Angles 4.2.11

Related Angles 4.2.12

Finding Exact Trigonometric Ratios State the ref angle and exact value of each ratio: CAST RULE 1. sin 1500 = 2. cos 2250 = 3. tan 3150 = 4. sin 2400 = 5. sin 6. cos 7. tan 8. tan 4.2.13

Finding Exact Trigonometric Ratios 1. sin 7500 = 2. cos -6750 = 3. cot 3150 = 4. csc 2400 = 5. csc 6. sec 7. cot 8. cot 4.2.14

Quadrantal Angles (0, 1) (-1, 0) (1, 0) (0, -1) 4.2.15

Finding Exact Trigonometric Ratios 1. sin 900 = 2. cos 900 = 3. sec 1800 = 4. csc 2700 = 5. csc 6. sec 7. cot 8. cot 4.2.16

Find Angle q ,Given an Exact Trigonometric Ratio Find the value of angle q. State the answers in degrees and radians. 00 ≤ q < 3600 0 ≤ q < 2p q = q = 4.2.17

Find Angle q , Given an Exact Trigonometric Ratio 4.2.18

Assignment Suggested Questions: Pages 199-201 A 1-4, 7, 11 , 13-23, 43 B 27-42, 44-49, 53 4.2.19