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Trigonometric Identities I. Pythagorean Identities A. sin 2 cos2 1 B. tan 2 1 sec 2 C. cot 2 1 csc 2 II. Sum and Difference of Angles Identities A. sin sin cos cos sin B. sin sin cos cos sin C. cos cos cos sin sin D. cos cos cos sin sin tan tan E. tan 1 tan tan tan tan F. tan 1 tan tan III. Double Angle Identities A. sin 2 2 sin cos B. cos2 cos 2 sin 2 = 2 cos2 1 = 1 2 sin 2 2 tan C. tan 2 1 tan 2 IV. Half Angle Identities 1 cos A. sin 2 2 1 cos B. cos 2 2 1 cos C. tan 2 1 cos 6-1 Inverse Trig Functions p. 468: 1-31 odd I. Inverse Trig Functions - y 2 2 1 B. y cos x if and only if x cos y where 0 y - y C. y tan 1 x if and only if x tan y where 2 2 A. y sin 1 x if and only if x sin y where Find the exact value of each expression 1 1. cos1 0 2. sin 1 2 3 4. cos 1 2 5. tan 1 1 3. tan 1 3 6. sin 1 1 Use a calculator to find each value. 3 7. sin 1 8. tan 1 5.1 7 9. cos1 Find the exact value of each expression. 3 10. sin sin 1 0.24 11. tan 1 tan 7 12. cos 1 cos 3 3 5 6-2 Inverse Trig Functions Continued p. 474:1-41 odd I. Inverse Trig Functions y 2 2 1 B. y sec x if and only if sec y x and 0 y A. y csc 1 x if and only if csc y x and C. y cot 1 x if and only if cot y x and 0 y Find the exact value of each expression. 1 3 2 1. cos sin 1 2. tan sin 2 2 4. csc cos 1 0 5. sin tan 1 4 2 6. cot 1 cot 3 Find the exact value of each. 2 3 8. sec 1 9. cot 1 1 3 Use a calculator to find each value. 11. cot 1 6.1 1 3. sec cos 1 2 10. csc 1 2 12. sec 1 4 4 7. cos 1 cos 3 Trigonometric Identities Trig Identities Worksheet: 1-6 all, 9, 13, 15, 19 I. Reciprocal Identities 1 1 A) sin D) csc csc sin 1 1 B) cos E) sec sec cos 1 1 C) tan F) cot cot tan II. Quotient Identities sin A) tan cos cos B) cot sin III. Pythagorean Identities A) sin 2 cos2 1 B) tan 2 1 sec 2 C) cot 2 1 csc 2 Simplify each expression. 1. csc 2 1 2. 1 sin x1 sin x 4. cos tan csc 5. 7. sin 2 x sin 2 x cot 2 x csc 1 cot 2 3. sin 2 csc 2 1 6. 1 1 2 sin tan 2 6-3 Trig Identities p. 480: 1-29 odd, omit 3 I. Guidelines for Verifying Trig Identities A. Begin with the most complicated side B. Work on one side only C. Rewrite sums or differences of quotients as one single quotient D. Rewrite in terms of sine and cosine only E. Factor ( GCF or Difference of Squares) F. Multiply ( Foil or Distributive Property) Verify each identity. 1. csc sin sin 2 cos2 2. sin cot tan sec 3. csc cot csc cot 1 4. csc 4 csc 2 cot 4 cot 2 5. csc x cot x 7. sin x 1 cos x 1 1 2 cot 2 1 sec 1 sec 6. 8 csc2 3 cot 2 3 5 csc2 6-4 Sum and Difference Formulas p. 491: 1-23 odd, 29, 31 I. Sum and Difference Identities A. cos B. cos C. sin D. sin E. tan F. tan Find the exact value of each expression. 1. cos 2. tan105 12 5. sin 40 cos 20 cos 40 sin 20 Find each of the following. a) sin b) tan 1 , 0 2 2 7 cos , 0 3 2 7. sin 8. Verify: cos cos 3. sin 13 12 4. csc15 6. cos100 cos 80 sin 100 sin 80 6-5 Double-Angle and Half-Angle Identities p. 501: 1-5 odd, 13-17 odd, 21, 51, 53 I. Double Angle Identities A) sin 2 2 sin cos B) cos 2 cos2 sin 2 2 cos2 1 1 2 sin 2 C) tan 2 2 tan 1 tan 2 II. Half Angle Identities 1 cos A) sin 2 2 B) cos C) tan 2 2 1 cos 2 1 cos 1 cos where + or – is determined by 2 Find the exact value of each expression. a) sin 2 1. sin b) cos2 12 , 0 13 2 c) sin 2 2. tan 15 3 , 8 2 Use the half angle identity to find each exact value. 5 3. sin 4. tan 22.5 8 Find the exact value. 3 5. sin 2 cos 1 2 5 6. cos sin 1 13 6-7 Trig Equations (I) p. 511: 1-29 odd Solve each equation for 0 2 . 3 3 1. sin 2. cot 2 3 4. sin 2 7. csc 2 2 3 2 3 2 3 5. 2 tan 1 1 8. cot 1 2 6 3. cos 1 2 6. 4 sin 2 3 0 6-8 Trig Equations (II) p. 518: 1-19 odd, 25 I. Solving Trig Equations A) Set trig function equal to a numerical value B) Apply trig identities to rename expression in terms of one trig function C) Factor: 1. GCF 2. Product of binomials D) Divide cos to produce tan Solve for 0 2 . 1. cos2 cos 0 2. 2 sin 2 3 sin 1 0 4. 2 cos2 sin 1 5. 7. cos 2 1 sin 3 sin cos 3. cot 1csc 1 0 6. sin 2 sin 0 8. sin tan 3 sin