SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. and θ is in quadrant IV. 1)

Transcription

1 Chapter 5-6 Review Math 116 Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use the fundamental identities to find the value of the trigonometric function. 1) Find sin θ if cos θ = and θ is in quadrant IV. 1) ) Find tan θ if sin θ = and s is in quadrant II. ) ) Find sin θ if sec θ = and tan θ < 0. ) ) Find csc θ if cot θ = - 5 and θ is in quadrant II. ) 5) Find sin θ if tan θ = - 5 and cos θ > 0. 5) 1 Complete the sentence so the result is an identity. Let x be any real number. 6) + sinx = 1 6) 7) sin x = ( )(cos x) 7) Write the first trigonometric function in terms of the second trigonometric function. 8) cot x; csc x 8) Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. 9) tan x(cot x - cos x) 9) 10) sin x - 1 cos (-x) 10) Use a graphing calculator to make a conjecture as to whether each equation is an identity. 11) sin x cos x = sin x 11) Perform the indicated operations and simplify the result. sin θ 1) 1 + sin θ - sin θ 1 - sin θ 1) 1) sec θ sin θ tan θ - 1 1) Factor the trigonometric expression and simplify. 1) secx - secx tanx + tanx 1) 1

2 Use the fundamental identities to simplify the expression. 15) cos θ - cos θ sinθ 15) 16) tanθ cscθ 16) Simplify the expression. 17) cot θ sec θ sin θ 17) 18) cos x sin x + cos x sec x 18) Verify that each equation is an identity. 19) csct - cos t sec t= cot t 19) 0) 1- sec θ tan θ + tan θ = - csc θ 0) 1 - sec θ 1) sec x - tan x = sec x + tan x 1) Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. sin θ + 1 ) = tan θ ) cos θ + cot θ Use Identities to find the exact value. ) cos -75 ) ) cos 1 ) Write in terms of the cofunction of a complementary angle. 5) cos 16 5) 6) cot 1 1 6) Use identities to fill in the blank with the appropriate trigonometric function name. 1 7) sin 59 = 1 7) Use the cofunction identities to find an angle θ that makes the statement true. 8) tan (θ - 10 ) = cot (θ + 5 ) 8) Use identities to write each expression as a function of θ. 9) cos (θ - ) 9)

3 Find the exact value of the expression using the provided information. 0) Find cos(s + t) given that cos s = 1, with s in quadrant I, and sin t = - 1, with t in quadrant 0) IV. 1) Find cos(s - t) given that cos s = - 1 1, with s in quadrant II, and sin t = 8, with t in 17 quadrant II. 1) Verify that the equation is an identity. ) sec + x = -csc x ) Find the exact value by using a sum or difference identity. ) sin 15 ) ) tan 75 ) 5) tan 105 5) 6) sin 7 1 6) Use trigonometric identities to find the exact value. 7) sin 5 cos 5 + cos 5 sin 5 7) 8) tan 10 + tan tan 10 tan 0 8) Use a sum or difference identity to find the exact value. 9) sin ) 0) tan ) Using a sum or difference identity, write the following as an expression involving functions of x. 1) sin - x 1) ) tan x - )

4 Find the exact value of the expression using the provided information. ) Find tan(s + t) given that sin s = 1, with s in quadrant II, and sin t = - 1, with t in ) quadrant IV. ) Find sin(s + t) given that cos s = - 1 6, with s in quadrant III, and cos t = -, with t in 5 ) quadrant III. Verify that the equation is an identity. 5) tan + x = -cot x 5) 6) sin - θ = -cos θ 6) Use an identity to write the expression as a single trigonometric function or as a single number. 7) cos.5-1 7) 8) sin.5 cos.5 8) 9) tan tan 15 9) 50) sin x cos x 50) Use identities to find the indicated value for each angle measure. 51) sin θ = 1, cos θ > 0 Find cos(θ). 51) 9 5) cos θ = 1, sin θ < 0 Find sin(θ). 5) 1 Verify that each equation is an identity. 5) cot(θ)= csc θ - cot θ 5) Find the exact value by using a half-angle identity. 5) sin.5 5) 55) cos 75 55) 56) tan )

5 Use an identity to write the expression as a single trigonometric function or as a single number. 57) 1 - cos 6 57) 58) 1 + cos 6 58) 59) sin 1 + cos 59) Establish the identity. 60) cos x - sin x = 1 - sin x 60) Find the exact value of the real number y. 61) y = sin-1 61) 6) y = csc-1(1) 6) Give the degree measure of θ. 6) θ = cos-1 6) 6) θ = cot-1 6) Use a calculator to give the value to the nearest degree. 65) θ = sin-1(.079) 65) 66) θ = cos-1(-.907) 66) Use a calculator to give the real number value. 67) y = tan-1(.577) 67) 68) y = arcsec (.8891) 68) 5

6 Graph the inverse circular function. 69) y = cos-1 x y 69) -1 1 x ) y = sin-1 x y 70) -1 1 x ) y = tan-1 x y 71) -1 1 x - - Evaluate the expression. 7) csc sin-1 5 7) 7) cos arcsin 1 7) 6

7 7) cos arcsin 5 + arccos 7) Use a calculator to find the value. Give answers as real numbers. 75) sin (arctan ) 75) 76) cos (cos-1(-.97)) 76) Write the following as an algebraic expression in u, u > 0. 77) cos(arcsin u) 77) 78) tan arcsec u + u 78) Solve the equation for the interval [0, ). 79) sinx = sin x 79) 80) cos x = sin x 80) 81) tan x + sec x = 1 81) Solve the equation in the interval [0, 60 ). 8) sinθ = 8) 8) sin θ = -sin θ 8) 8) cosθ + 7 sin θ = 5 8) Determine the solution set of each equation in radians (for x) or degrees (for θ) to the nearest tenth as appropriate. 85) sin x - 1 = 0 85) 86) sin x + sin x = 1 86) 87) tan θ 5 - tan θ = 1 87) Solve the equation for solutions in the interval [0, ). 88) sin x = 88) 89) sin x cos x = 1 89) 90) csc x = 0 90) 7

8 91) cos x = 1 91) Solve the equation for solutions in the interval [0, 60 ). Round to the nearest degree. 9) sin θ = cos θ 9) 9) sec θ = 9) 9) tan θ = 5 9) 95) cot θ = 1 95) Solve the equation for x. 96) y = 7 sin x 96) 97) y = cot x 97) 98) y = tan x ) Solve the equation for exact solutions. 99) 6 cos-1 x = 99) 100) cos-1 x = sin ) Solve the equation. 101) arccos x + arccos x = arccos 1 101) 10) arcsin x + arctan x = 10) 8

9 Answer Key Testname: TRIG CH. 5 6 REVIEW 1) - 5 ) ) ) ) ) cosx 7) tan x 8) ± csc x - 1 9) 1 - sin x 10) -cos x 11) Identity 1) - tanθ 1) 0 1) 1 15) cosθ 16) secθ 17) 1 18) csc x 19) csct - cos t sec t = csct - cos t 1 cos t = csc t - 1 = cott 0) 1- sec θ tan θ + tan θ 1 - sec θ = (1 - sec θ) + tanθ = 1 - sec θ + sec θ + tanθ = sec θ - sec θ tan θ(1 - sec θ) tan θ(1 - sec θ) tan θ(1 - sec θ) = sec θ( sec θ - 1) tan θ(1 - sec θ) = - sec θ tan θ = - cos θ cos θ sin θ = - sin θ = - csc θ 1) sec x - tan x = (sec x + tan x)(sec x - tan x) = (sec x + tan x)(1) = sec x + tan x. ) Identity ) ) ) sin ) tan - 7 7) sec 8) θ = 75 9) -cos θ 0) + 6 9

10 Answer Key Testname: TRIG CH. 5 6 REVIEW 1) 0 1 ) sec + x = 1 cos (/) cos x - sin (/) sin x = 1 = -csc x. 0 cos x - 1 sin x ) 6 - ) + 5) - - 6) + 6 7) 8) 9) - 6-0) - + 1) cos x - sin x ) tan x tan x ) ) ) tan 6) sin sin ((/) + x) sin (/) cos x + sin x cos (/) + x = = cos ((/) + x) cos (/) cos x - sin (/) sin x - θ = sin cos θ - cos = 1 cos x + sin x 0 0 cos x - 1 sin x sin θ = (-1) cos θ - 0 sin θ = - cos θ = -cot x. 7) 8) 9) 50) sin x 51) )

11 Answer Key Testname: TRIG CH. 5 6 REVIEW 5) cot(θ) = cos(θ) sin(θ) = 1 - sin θ sin θ cos θ = 1 sinθ - cos θ sin θ = csc θ - cot θ 5) 1 55) ) ) sin 58) cos 59) tan 17 60) cos x - sin x = 1 + cos x cos x = 1 + cos x cos x 1 - cos x cos x = cos x = 1 - sin x = 1 - sin x = 1 - sin x 61) 6) 6) 0 6) 60 65) 1 66) 11 67) ) ) y -1 1 x

12 Answer Key Testname: TRIG CH. 5 6 REVIEW 70) y -1 1 x - 71) - y -1 1 x - - 7) 5 7) 7 8 7) ).89 76) ) 1 - u 78) u 79) 0,, 6, ), 5 81) {0} 8) {60, 10, 0, 00 } 8) {0, 10, 180, 0 } 8) {90, 8.6, 11. } 85) 6 + n, n 1

13 Answer Key Testname: TRIG CH. 5 6 REVIEW 86) 6 + n, n, + n 87) { n, n} 88) 1, 6,, 7 1, 7 6, 1 1, 5, ), 5 90) 91) 8, 9 8, 7 8, ) {0, 90, 150, 70 } 9) {15, 165, 195, 5 } 9) {, 57, 1, 17, 1, 7, 0, 7 } 95) {15 } 96) x = arcsin y 7 97) x = arccot y 98) x = 1 arctan y ) 100) ) x = 1 10) x = 1 1