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MATH 4H TRIGONOMETRY 2 HOMEWORK NAME__________________________ DATE___________________________ HW # 50: Inverse Trigonometric Functions (Packet pp. 5 – 6) – ALL HW # 51: Finish Evaluating Using Inverse Trig Functions (Packet p. 7) Solving Linear Equations (Packet p. 8) – ALL HW # 52: Solving First Degree Equations (Packet p. 10) – ALL HW # 53: Second Degree Trigonometric Equations (Packet p. 12) - # 1, 4, 5 HW # 54: Quadratic Trig Equations (Packet p. 13) - # 2, 3, 4, 6 HW # 55: Trigonometric Identities (Packet p. 15) - # 1, 3, 8 Proving Trigonometric Identities (Packet p. 16) - # 4 HW # 56: Trigonometric Identities (Packet p. 15) - # 4 – 7 Sum and Difference Identities (Packet p. 18) - # 1, 11 HW # 57: Using Pythagorean Identities to Solve Quadratic Equations (Packet p. 19) - # 3, 5, 7 HW # 58: Trigonometry Trig 2 Test Review (Packet pp. 21 – 22) Study for Test!!! Inverse Trigonometric Graphs On the axes below, sketch a graph of sin on the interval 2 2 . sin x y On the axes below, sketch a graph of . State the range and domian of this function. ________________________________ y x 1 On the axes below, sketch a graph of cos on the interval 2 2 . cos x y On the axes below, sketch a graph of . State the range and domian of this function. ________________________________ y x 2 On the axes below, sketch a graph of tan on the interval 2 2 . tan x y On the axes below, sketch a graph of . State the range and domian of this function. ________________________________ y x 3 Domain Restrictions on Inverse Trigonometric Functions 4 Inverse Trigonometric Functions 1) 2) 5 6 Evaluating Using Inverse Trig Functions 1. If cos 2. If sin √ , what is the measure of angle ? , find ∠ . 3. What is the smallest positive value of 4. If tan 6. If cos (1) 30° 1 , find (2) 60° 7. The value of cos √ (3) sin 11. cos (2) 13. sin tan 1 17. sin sin 1 18. If 3 sin √ (3) 30° is (4) 60° 12. cos 14. 2 16. sin tan 1 sin 1 sin cos 1 1, then which is an expression for (1) sin 3 (2) sin 19. If 5 sin tan 1 ? tan 1 15. tan ∠ . (4) 150° √ sin , find (4) , the value of cot (1) √ , what is the value of tan ? 9. What is the value of sin 10. If sin measures (3) 120° (2) 8. If 5. If 1 is sin (1) 1 ∠ . , then cos ? that satisfies (3) 3 3, express in inverse trigonometric form? sin (4) sin as an inverse of a trigonometric equation. 7 Solving Linear Equations 1. 2 1 0 3. 4 1 2 5. 3 2 7. 3 √3 9. 11. 3 12 5 1 2 2 7 2. 3 √3 0 4. 5 1 5 6. √2 2√2 8. 5 3 11 10. √2 9 12. 4 8 3 1 5 √ 2 2 Linear Trigonometric Equations Find the exact solution set of each equation if ° 1. 2 1 0 3. 4 1 2 5. 3 2 1 2 7 Find the exact values for 7. 3 9. 11. 3 12 5 2 2. 3 √3 0 4. 5 1 5 √2 6. in the interval √3 11 9 °. 2√2 . 8. 5 3 10. √2 12. 4 9 3 1 5 √ 2 2 Solving First Degree Equations 1-5, solve for in the interval 0 360 . (Express your answer to the nearest degree) 1. 2sin 3 0 R: Q: A: To Find an Angle in Quadrant: I A In Degrees (Reference Angle of θ) II S 180 III T 180 IV C 360 2. 3cos 1 1 3. 3tan 2 tan 4. 2(sin 1) sin 3 5. 10(cos 1) 30 6-7, solve for in the interval 0 2 . 6. sec 6 3sec 7. 3(sin 2) 1 5sin 10 Solving First Degree Equations Continued Directions: In 1-3, solve for θ in the interval 0 360 . (Express your answer to the nearest degree) In 4-6, solve for θ in the interval 0 2 . *Hint Multiply Answer by 180 4. 2(sin 2) 2 1. 2 tan 3 5 2. 4(csc 2) csc 14 5. 3sec 2 (3sec 3) 3 6. 4 cos 3 3 3 3. 6 cot 5cot 2 3 2 11 Second Degree Trigonometric Equations Remember: If you have difficulties factoring, you can use the quadratic formula. The Quadratic Formula b b 2 4ac x 2a Solve for θ on the interval 0 360 . 2. tan (tan 1) tan 3 1. 2 cos cos 1 2 Solve for θ on the interval 0 2 . 3. cos 5. 1 cos 2 3 0 12 4. 2 3 1 0 6. 8 2 1 0 Quadratic Trig Equations Find the exact solution set of each equation if ° 1. 3 0 2. 2 3. 2 6. 2 1 0 Find the exact values for 4. °. 1 0 in the interval . 5. 0 13 0 Second Degree Trigonometric Equations 2 cos 0 when 90 180 . 1. Solve for all values of cos 2. 2 What is the value of θ in the interval 0 x 360 that satisfies the equation cos x 3sin x 1 ? 3. Solve the equation sin x 2 cos x 2 on the interval 0 x 90 . 4. Find, to the nearest degree, all values of θ in the interval 0 180 that satisfy the equation 4 tan 2 3 tan 2 0 . 5. Find all values of x in the interval 0 x 360 that satisfy the equation 2 sin 2 x sin x 2 3 Express your answers to the nearest degree. 2 6. Solve the equation tan x sin x 2 tan x in the interval 0 2 14 2 . Trigonometric Identities Pythagorean Identities Double Angles 1 sin 2 cos 2 cos 2 cos 2 1 1 2 sin cos 2 1 1 2 2 Notes: 1. If 2. If 3. If 1 1 1, then ______________________________________ and ______________________________________ , then _____________________________________ , then _______________________________________ Express in simplest form: 1) 2) 3) 4) 5) 6) 7) sin 2 9) 1 cos 2 8) 15 Proving Trigonometric Identities Ex. Prove each identity 1. sin 2 sec 2 sin 2. sin sin 2 2 cos 3. sin 2 tan 1 cos 2 4. sin 2 csc 2 cos 5. cos 2 cot sin sin sec 6. tan cot 16 2 sin 2 7. sin 2 2 tan 1 tan 2 2 8. sin 2 sec 2 tan 2 9. cos sin 1 sin 2 2 10. cos sin 1 sin 2 2 sin 2 cot sec csc 11. sin 2 12. 17 cos 2 sin csc sin sin Sum and Difference Identities sin sin cos cos sin cos cos cos sin sin sin sin cos cos sin cos cos cos sin sin Use the angle sum identity to find the exact value of each. 1) cos 105° 2) sin 195° 3) cos 195° 4) cos 165° 5) cos 285° 6) cos 255° Use the angle difference identity to find the exact value of each. 11) cos 75° 12) cos −15° 13) tan 75° 14) cos 15° 15) tan −105° 16) sin 105° 18 Using Pythagorean Identities to Solve Quadratic Equations Solve the following equations; 2 1 2 ∈ , 2 2) 2 cos x 3 sin x 3 0 0 2 3) 3 cos x 5 sin x 4 2 5) 2 7) 2 9) 3 2 4 4) 2 cos 6) 2 3 8) 2 3 10) 19 3 2 1 3 2 0 Using Double Angle Identities to Solve Quadratic Equations 1 2 3) 2 5) 2 7) 2 9) 2 3 1 2) 3 0 0 2 2 4) 2 0 6) 2 0 2 8) 1 10) 20 2 0 2 Trigonometry Test2 Review 1. Graph 2. Graph 3. Graph 4. Which of the following are not an inverse trigonometric functions? 5. Use the angle sum identity to find the exact value of each. a) 75° b) 105° 6. Use the angle difference identity to find the exact value of each. 75° a) 15° b) 7. Simplify the following expressions: a) 2 b) 21 8. Prove the identity Solve the following equations for x in the interval 0 2 9. 4 2 2 10. 2 3 0 11. 2 1 0 13. 2 3 3 12. 2 0 15. 2 2 0 14. 22 1 0 0 23 Trigonometric Identities Reciprocal Identities sin cos csc sec sin csc 1 1 cos cos sec Pythagorean Identities 1 Ratio Identities tan tan cot cot tan cot 1 1 csc Double Angles 2 sin cos cos 2 cos 2 2 cos 2 1 1 Cofunctions Identities 1 sin 2 cos sin cos 90° cos sin 90° tan cot 90° sec 90° sec csc 90° cot tan 90° 1 2 Sum & Difference Identities sin sin cos cos sin cos cos cos sin sin sin sin cos cos sin cos cos cos sin sin 24