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Name: ________________________ Class: ___________________ Date: __________ ID: A Review 2 Show All the steps to get full credit: 1. Verify the identity shown below. cos θ (tan θ − sec θ ) = sin θ 1 − csc θ 2. Verify the identity shown below. sec θ − sin θ tan θ = cos θ 3. Verify the identity shown below. ÊÁ 2 ˆ ÁÁ tan θ − 1 ˜˜˜ cot θ ÁË ˜¯ = sec θ + csc θ sin θ − cos θ 4. Verify the identity shown below. cos α − cos β sin α − sin β + =0 sin α + sin β cos α + cos β 5. Verify the identity shown below. ˆ˜ Áπ 2 2Ê sec µ − cot ÁÁÁÁ − µ ˜˜˜ = 1 ˜¯ Ë2 6. Verify the given identity. 2 2 cos(x + y) cos(x − y) = cos x − sin y 7. If sinx = 3 1 and cos x = , evaluate the following function. 2 2 tanx 8. Use fundamental identities to simplify the expression below and then determine which of the following is not equivalent. sin α (csc α − sin α ) 1 Name: ________________________ ID: A 9. Factor; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. cot 2 α tan2 α + cot 2 α 10. Multiply; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. (2 − 2 cos x ) (2 + 2 cos x ) 11. If x = 10 cosθ , use trigonometric substitution to write where 0 < θ < π . 12. If x = 6 sinθ , use trigonometric substitution to write where 0 < θ < π 2 100 − x as a trigonometric function of θ, 2 36 − x as a trigonometric function of θ, 2 . 13. Solve the given equation sec x − 2 = 0 14. Solve the following equation. 2 sinx − 1 = 0 15. Solve the following equation. sin2 x + sinx = 0 16. Solve the multiple-angle equation. cos 3 x =− 2 2 ÍÈ ÍÎ 2 utility to approximate the angle x. Round answers to three decimal places. Í π 17. Use the Quadratic Formula to solve the given equation on the interval ÍÍÍÍ 0, 18 tan2 x − 13 tanx + 2 = 0 2 ˆ˜ ˜˜ ; then use a graphing ˜˜ ¯ Name: ________________________ ID: A 18. Use the identities where needed to find all solutions (if they exist) of the given equation on the interval [0, 2π ) . cos 2 x − 5 sinx + 5 = 0 19. The horizontal distance d (in feet) traveled by a projectile with an initial speed of v feet per second is modeled by d= 2 v sin2θ , 32 where θ is the angle at which the projectile is launched. Find the horizontal distance traveled by a golf ball that is hit with an initial speed of 50 feet per second when the ball is hit at an angle of θ = 50° . Round to the nearest foot. 20. Find the exact value of the given expression. ÁÊ 5π 5π ˜ˆ˜ ˜ sinÁÁÁÁ − 4 ˜˜¯ Ë 3 21. Find the exact value of the given expression. cos (240° + 225° ) 22. Find the exact value of the given expression using a sum or difference formula. sin285° 23. Write the given expression as the sine of an angle. sin105° cos 35° + sin35° cos 105° 24. Write the given expression as the tangent of an angle. tan6x + tan2x 1 − tan6x tan2x 25. Find the exact value of sin(u + v ) given that sinu = 7 12 and cos v = − . (Both u and v are in 25 13 Quadrant II.) 26. Find the exact value of cos (u + v ) given that sinu = Quadrant II.) 1 7 12 and cos v = − . (Both u and v are in 25 13 Name: ________________________ ID: A 27. Write the given expression as an algebraic expression. sin(arcsinx + arccos x) 28. Simplify the given expression algebraically. ÊÁ π ˆ˜ sinÁÁÁ x − ˜˜˜˜ ÁË 2¯ 29. Use the figure below to determine the exact value of the given function. csc 2θ 30. Find the exact solutions of the given equation in the interval [0, 2π ) . cos 2x + 3 cos x + 2 = 0 31. Use a double angle formula to rewrite the given expression. 6 cos 2 x − 3 32. Use a double-angle formula to find the exact value of cos 2u when sinu = 7 π , where <u<π. 25 2 33. Use the figure below to find the exact value of the given trigonometric expression. sin θ 2 15 36 (figure not necessarily to scale) 4 Name: ________________________ ID: A 34. Use the half-angle formulas to determine the exact value of the given trigonometric expression. tan 3π 8 35. Use the half-angle formula to simplify the given expression. 1 + cos 4x 2 36. Use the product-to-sum formula to write the given product as a sum or difference. 12 sin π 6 cos π 6 37. Use the sum-to-product formulas to write the given expression as a product. sin6θ − sin4θ 38. The range of a projectile fired at an angle θ with the horizontal and with an initial velocity of v 0 feet 1 2 v sin2θ where r is measured in feet. A golfer strikes a golf ball at 120 feet per 32 0 second. Ignoring the effects of air resistance, at what angle must the golfer hit the ball so that it travels 140 feet? (Round answer to nearest angle.) per second is r = 5 ID: A Review 2 Answer Section SHORT ANSWER 1. ANS: PTS: 1 2. ANS: OBJ: Verify identities sec θ − sin θ tan θ = 1 sin θ − sin θ ⋅ cos θ cos θ 1 sin2 θ = − cos θ cos θ = 1 − sin θ cos θ = cos θ cos θ 2 2 = cos θ PTS: 1 OBJ: Verify identities 1 ID: A 3. ANS: PTS: 1 OBJ: Verify identities 2 ID: A 4. ANS: PTS: 1 OBJ: Verify identities 3 ID: A 5. ANS: PTS: 1 6. ANS: OBJ: Verify identities PTS: 1 7. ANS: OBJ: Use sum and difference formulas to verify trig identities tanx = 3 3 PTS: 1 8. ANS: OBJ: Evaluate trig function given other trig values 1 − cot 2 α PTS: 1 OBJ: Use fundamental identities to determine equivalent expression 4 ID: A 9. ANS: sec 2 α PTS: 1 10. ANS: OBJ: Use fundamental identities to determine equivalent expression 4 − cos 2 x PTS: 1 11. ANS: OBJ: Use fundamental identities to determine equivalent expression PTS: 1 12. ANS: OBJ: Write algebraic expressions as trig functions with trig substitution PTS: 1 13. ANS: OBJ: Write algebraic expressions as trig functions with trig substitution 10 sinθ 6 cosθ x= 5π 3 PTS: 1 14. ANS: x= π 6 + 2nπ and x = PTS: 1 15. ANS: x = nπ and x = PTS: 1 16. ANS: x= OBJ: Verify solutions to trig equations 5π + 2nπ , where n is an integer 6 OBJ: Solve trig equations 3π + nπ , where n is an integer 4 OBJ: Solve trig equations 5π 7π + 4nπ and + 4nπ , where n is an integer 3 3 PTS: 1 17. ANS: OBJ: Solve multiple-angle equations PTS: 1 18. ANS: OBJ: Use the Quadratic Formula to solve trig equations x = 0.219, 0.464 x= π 2 PTS: 1 OBJ: Solve trig equations by factoring 5 ID: A 19. ANS: 77 PTS: 1 20. ANS: 3 +1 2 2 PTS: 1 21. ANS: 1− OBJ: Find exact value of expression using sum formula 3 2 2 PTS: 1 22. ANS: OBJ: Find exact value of expression using sum formula − 3 −1 2 2 PTS: 1 23. ANS: OBJ: Find exact value of expression using sum or difference formula sin(140° ) PTS: 1 24. ANS: OBJ: Rewrite an expression using a sum or difference formula tan(8x ) PTS: 1 25. ANS: OBJ: Rewrite an expression using a sum or difference formula sin(u + v ) = − 204 325 PTS: 1 26. ANS: cos (u + v ) = PTS: 1 27. ANS: OBJ: Find exact value of expression using sum or difference formula with constraints 253 325 OBJ: Find exact value of expression using sum or difference formula with constraints 1 PTS: 1 28. ANS: OBJ: Write trig expressions as algebraic expressions −cos x PTS: 1 OBJ: Simplify trig expressions using addition and subtraction formulas 6 ID: A 29. ANS: csc 2θ = 13 12 PTS: 1 30. ANS: x= OBJ: Find exact value of trig function from diagram 2π 4π , π, 3 3 PTS: 1 31. ANS: OBJ: Find exact solutions to trig equations involving multiple angles 3 cos 2x PTS: 1 32. ANS: cos 2u = OBJ: Rewrite an expression as a double angle 527 625 PTS: 1 33. ANS: OBJ: Find exact value of double angle given quadrant restraints 26 26 PTS: 1 34. ANS: tan 3π = 8 OBJ: Find exact value of expression using the half-angle formula 2 +1 PTS: 1 35. ANS: OBJ: Find exact value of expression using the half-angle formula cos 2x PTS: 1 36. ANS: 6 sin OBJ: Rewrite an expression with a product-to-sum formula π 3 PTS: 1 37. ANS: OBJ: Use the product-to-sum formula to rewrite products PTS: 1 38. ANS: OBJ: Use the sum-to-product formula to rewrite difference as a product 2 cos 5θ sin θ 9° PTS: 1 OBJ: Solve problems dealing with multiple angle formula 7