HW#50: Finish Evaluating Using Inverse Trig Functions (Packet p. 7) Solving Linear Equations (Packet p. 8) ALL

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1 MATH 4R TRIGONOMETRY HOMEWORK NAME DATE HW#49: Inverse Trigonometric Functions (Packet pp. 5 6) ALL HW#50: Finish Evaluating Using Inverse Trig Functions (Packet p. 7) Solving Linear Equations (Packet p. 8) ALL HW#51: Linear Trigonometric Equations (Packet p. 9) Even Problems HW#5: Solving First Degree Equations (Packet p. 10) ALL HW#53: Second Degree Trigonometric Equations (Packet p. 1) - # 1, 4, 5 HW#54: Quadratic Trig Equations (Packet p. 13) - #, 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 Test Review (Packet pp. 1 ) Study for Test!!!

2 Inverse Trigonometric Graphs On the axes below, sketch a graph of y = sin(x) on the interval π x π. y = sin (x) x y On the axes below, sketch a graph of y = sin 1 (x). State the range and domian of this function. y = sin 1 (x) y x 1

3 On the axes below, sketch a graph of y = cos(x) on the interval π x π. y = cos (x) x y On the axes below, sketch a graph of y = cos 1 (x). State the range and domian of this function. y = cos 1 (x) y x

4 On the axes below, sketch a graph of y = tan(x) on the interval π x π. y = tan (x) x y On the axes below, sketch a graph of y = tan 1 (x). State the range and domian of this function. y = tan 1 (x) y x 3

5 Domain Restrictions on Inverse Trigonometric Functions 4

6 Inverse Trigonometric Functions 1) ) 5

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8 Evaluating Using Inverse Trig Functions 1. If θ = Arc cos 3, what is the measure of angle θ?. If x = Arc sin ( 1 ), find m x. 3. What is the smallest positive value of x that satisfies x = Arc cos 1? 4. If θ = Arc tan( 1), find m θ. 5. If x = Arc sin ( 1 ), find m x. 6. If x = arc cos ( 1 ), then x measures (1) 30 () 60 (3) 10 (4) The value of Arc sin( 1) is (1) 1 () π (3) π (4) π 4 8. If θ = Arc cos, what is the value of tan θ? 9. What is the value of sin(arc tan 1)? 10. If y = sin (Arc cot 1 ), the value of y is (1) 1 () 3 (3) 30 (4) cos (Arc sin 3 ) = 1. cos(arc tan 1) = 13. Arc sin 1 + Arc tan 1 = 14. (Arc sin 1) = 15. tan(arc tan 1) = 16. sin (Arc sin 1 ) = 17. sin(arc sin 1 + Arc cos 1) = 18. If 3 sin x = 1, then which is an expression for x in inverse trigonometric form? (1) x = Arc sin 3 (3) x = Arc sin ( 1 3 ) () x = Arc sin( 3) (4) x = Arc sin ( 1 3 ) 19. If 5 sin A = 3, express A as an inverse of a trigonometric equation. 7

9 Solving Linear Equations 1. x 1 = 0. 3x + 3 = x 1 = x (x + 1) = (x ) = x 7 6. x + = 7. 3x 3 = x 8. 5x + 3 = 3x x + 1 = x x + = 11. 3x + 5 = x (x + 1) = (x + ) 8

10 Linear Trigonometric Equations Find the exact solution set of each equation if 0 θ < cosθ 1 = 0. 3tanθ + 3 = sinθ 1 = sinθ (cosθ + 1) = (tanθ ) = tanθ 7 6. secθ + = Find the exact values for θ in the interval 0 θ π. 7. 3sinθ 3 = sinθ 8. 5cosθ + 3 = 3cosθ tanθ + 1 = tanθ sinθ + = 11. 3cscθ + 5 = cscθ (cotθ + 1) = (cotθ + ) 9

11 Solving First Degree Equations 1-5, solve for in the interval (Express your answer to the nearest degree) 1. sin 3 0 R: Q: A: To Find an Angle in Quadrant: I A II III S T In Degrees (Reference Angle of θ) IV C cos tan tan 4. (sin 1) sin (cos 1) , solve for in the interval sec 6 3sec 7. 3(sin ) 1 5sin 10

12 Solving First Degree Equations Continued Directions: In 1-3, solve for θ in the interval In 4-6, solve for θ in the interval 0. (Express your answer to the nearest degree) *Hint Multiply Answer by 1. tan (sin ) (csc ) csc sec (3sec 3) cot 5cot cos

13 Second Degree Trigonometric Equations Remember: If you have difficulties factoring, you can use the quadratic formula. Solve for θ on the interval The Quadratic Formula x b b 4ac a 1.. tan (tan 1) tan 3 cos cos 1 Solve for θ on the interval cos 1 cos 4. sin θ 3sinθ + 1 = 0 5. tan θ tan θ 3 = cos θ cosθ 1 = 0 1

14 Quadratic Trig Equations Find the exact solution set of each equation if 0 θ < tan θ 3 = 0. sin θ + sinθ 1 = 0 3. sinθcosθ + cosθ = 0 Find the exact values for θ in the interval 0 θ π. 4. tan θ 1 = 0 5. cos 3 θ = cosθ 6. sinθ cscθ = 0 13

15 Second Degree Trigonometric Equations 1. Solve for all values of cos cos 0 when 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 cos x on the interval 0 x Find, to the nearest degree, all values of θ in the interval that satisfy the equation 4tan 3tan Find all values of x in the interval 0 x 360 that satisfy the equation sin x sin x 3 Express your answers to the nearest degree. 6. Solve the equation tan xsin x tan x in the interval 0 θ π. 14

16 Trigonometric Identities Pythagorean Identities Double Angles Notes: sin θ + cos θ = 1 sin θ = sin θ cos θ tan θ + 1 = sec θ cos θ = cos θ sin θ 1 + cot θ = csc θ cos θ = cos θ 1 cos θ = 1 sin θ tanθ = tanθ 1 tan θ 1. If sin θ + cos θ = 1, then sin θ = and cos θ =. If tan θ + 1 = sec θ, then tan θ = 3. If 1 + cot θ = csc θ, then cot θ = Express in simplest form: 1) 1 cos x sin x ) sin x cos x 3) sinθ sin θ 4) sinxcos x sinx 5) csc x 1 cscx 1 6) cosθ(secθ cosθ) 7) sinθ + cotθcosθ 8) sec xtan x + sec x 9) sin 1 cos 15

17 Proving Trigonometric Identities Ex. Prove each identity sin sec sin. 1. sin sin cos sin 3. tan 1 cos 4. sin csc cos 5. cos cot sin 6. sin sec tan cot sin 16

18 7. sin tan 8. sin sec tan 1 tan 9. cos sin 1 sin 10. cos sin 1 sin cos sin 11. cot sec csc 1. sin csc sin sin sin 17

19 Sum and Difference Identities sin(a + B) = sin A cos B + cos A sin B cos(a + B) = cos A cos B sin A sin B sin(a B) = sin A cos B cos A sin B cos(a B) = cos A cos B + sin A sin B Use the angle sum identity to find the exact value of each. 1) cos 105 ) sin 195 3) cos 195 4) cos 165 5) cos 85 6) cos 55 Use the angle difference identity to find the exact value of each. 11) cos 75 1) cos 15 13) tan 75 14) cos 15 15) tan ) sin

20 Using Pythagorean Identities to Solve Quadratic Equations Solve the following equations; x [0, π] 1) sin x cosx = 0 ) cos x 3sin x 3 0 3) 3cos x 5sin x 4 4) sin x cos x = 1 5) sin x = + cosx 6) sin x + 3cosx 3 = 0 7) sec x tanx = 4 8) cscx 3cot x = 9) 3sinx cos x = 3 10) cos x = sinx 19

21 Using Double Angle Identities to Solve Quadratic Equations 1) cosa = 3sinA 1 ) 3cosθ = sinθ 3) sinx sinx = 0 4) cosx cosx = 0 5) sinxcosx = sinx 6) tanx tanx = 0 7) cosx cosx = 0 8) cos x cosx = 0 9) cosx 1 = sin x 10) sin x = cosx 0

Use the angle sum identity to find the exact value of each. a) sin 75 b) cos 105 6.

22 Trigonometry Test Review 1. Graph y = sin 1 x. Graph y = cos 1 x 3. Graph y = tan 1 x 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) sin 75 b) cos Use the angle difference identity to find the exact value of each. a) sin 15 b) tan Simplify the following expressions: a) sinθsecθ b) cosx cos x 1

23 8. Prove the identity sinxtanx + cosx = secx Solve the following equations for x in the interval 0 x π 9. 4tanx + = tanx 10. cosx + 3 = sinx 1 = 0 1. sin x sinx 1 = sin x + 3cosx 3 = sin x cos x = sin x cosx = 0

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25 Trigonometric Identities Reciprocal Identities Ratio Identities sin θ = 1 csc θ cos θ = 1 secθ tan θ = 1 cot θ tan θ = sin θ cos θ csc θ = 1 sin θ sec θ = 1 cos θ cot θ = 1 tan θ cot θ = cos θ sinθ sin θ csc θ = 1 cos θ sec θ = 1 tan θ cot θ = 1 Pythagorean Identities sin θ + cos θ = 1 tan θ + 1 = sec θ Cofunctions Identities sin θ = cos(90 θ) cos θ = sin(90 θ) 1 + cot θ = csc θ tan θ = cot(90 θ) Double Angles sin θ = sin θ cos θ cos θ = cos θ sin θ csc θ = sec(90 θ) sec θ = csc(90 θ) cot θ = tan(90 θ) cos θ = cos θ 1 cos θ = 1 sin θ Sum & Difference Identities sin(a + B) = sin A cos B + cos A sin B cos(a + B) = cos A cos B sin A sin B sin(a B) = sin A cos B cos A sin B cos(a B) = cos A cos B + sin A sin B 4

Pre Calculus Worksheet: Fundamental Identities Day 1

Pre Calculus Worksheet: Fundamental Identities Day 1 Use the indicated strategy from your notes to simplify each expression. Each section may use the indicated strategy AND those strategies before. Strategy