Ch 5 and 6 Exam Review Note: These are only a sample of the type of exerices that may appear on the exam. Anything covered in class or in homework may appear on the exam. 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. ) Find cot θ if tan θ = 7 and θ is in quadrant III. ) Find csc θ if cot θ = - 15 and θ is in quadrant II. ) Find cos θ if tan θ = and sin θ < 0. Complete the sentence so the result is an identity. Let x be any real number. 5) 1 - = sin x 6) sin x = ( )(cos x) Perform the transformation. 7) Write sec x in terms of tan x. Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. 8) (1 + cot θ)(1 - cot θ) - csc θ 9) tan θ csc θ Perform the indicated operations and simplify the result so there are no quotients. 10) sec θ - 1 sec θ 11) sin θ cos θ + cos θ sin θ 1) sec θ sin θ tan θ - 1 Factor the trigonometric expression and simplify. 1) sec x - sec x tan x + tan x 1) 1 - sin x 15) 1 - sin x + sin x 1
Use the fundamental identities to simplify the expression. 16) cos θ sin θ + csc θ sin θ 17) tan θ cot θ 18) sin θ cos θ sec θ csc θ Verify that each equation is an identity. 19) tan θ csc θ = sec θ 0) (1 - cos x)(1 + cos x) = sin x 1) sec θ - 1 tan θ = tan θ sec θ + 1 ) 1 - cot θ 1 + cot θ + 1 = sin θ ) sin x cos x = sin x (cos x - cos x) Decide whether the expression is or is not an identity. ) sin x + cos x = 1 5) sec x - tan x = 0 Use Identities to find the exact value. 6) cos 165 7) cos - 7" 1 Use identities to write each expression as a function of θ. 8) cos (θ + 90 ) Find the exact value of the expression using the provided information. 9) Find cos(s + t) given that cos s = 1, with s in quadrant I, and sin t = 1, with t in quadrant II. 0) Find cos(s - t) given that sin s = -, with s in quadrant IV, and sin t = - 5, with t in quadrant IV. 6
Write in terms of the cofunction of a complementary angle. 1) sec 1 18 Use the cofunction identities to find an angle θ that makes the statement true. ) sec (6θ + 17 ) = csc (θ - 7 ) Use identities to fill in the blank with the appropriate trigonometric function name. ) 6 = tan 6 Verify that the equation is an identity. ) cos x + " 6 = cos x - 1 sin x 5) cos(x - y) - cos(x + y) = sin x sin y Use a sum or difference identity to find the exact value. 6) tan(- 15 ) 7) tan 170 - tan 50 1 + tan 170 tan 50 Find the exact value of the expression using the provided information. 8) Find tan(s + t) given that cos s = 1, with s in quadrant I, and sin t = - 1, with t in quadrant IV. Using a sum or difference identity, write the following as an expression involving functions of x. 9) tan (x + ") Verify that the equation is an identity. 0) sin " - θ = -cos θ Use an identity to write the expression as a single trigonometric function or as a single number. tan 0 1) 1 - tan 0 ) cos x - sin x Use identities to find the indicated value for each angle measure. ) sin θ = 0, cos θ > 0 Find cos(θ). 9 ) cos θ = - 5 1, " < θ < " Find cos(θ).
Express the function as a trigonometric function of x. 5) sin x Verify that each equation is an identity. 6) cos(x) = cos x - sin x cos x Write the product as a sum or difference of trigonometric functions. 7) cos 8x cos x 8) 10 cos 6 cos 1 Rewrite the following as a product of trigonometric functions. 9) cos " 11 - cos " 50) cos 9 - cos Find the exact value by using a half-angle identity. 51) sin 75 5) cos 165 Determine all solutions of the equation in radians. 5) Find sin θ, given that cos θ = 1 and θ terminates in 0 < θ < 90. 5) Find tan x, given that tan x = - and x terminates in 90 < x < 180. Verify that the equation is an identity. 55) sec u = sec u sec u + 1 Find the exact value of the real number y. 56) y = csc -1 () 57) y = sin -1 58) y = csc -1 (-1) 59) y = sin -1 1 Use a calculator to give the value to the nearest degree. 60) θ = tan -1 (0.577)
Use a calculator to give the real number value. Round the answer to 7 decimal places. 61) y = cos -1 (-0.997) Give the exact value of the expression. 6) sin (arctan ) 6) csc(csc -1 ) 6) cos arcsin 5 1 + arccos 5 Write the following as an algebraic expression in u, u > 0. 65) cos(arctan u) 66) tan arcsec u + 5 u Solve the equation for exact solutions over the interval [0, "). 67) sin x = sin x 68) tan x + sec x = 1 Solve the equation in the interval [0, 60 ). Give solutions to the nearest tenth, if necessary. 69) sin θ - sin θ - = 0 Solve the equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. 70) cos x + cos x = -1 Solve the equation in the interval [0, 60 ). Give solutions to the nearest tenth, if necessary. 71) sin θ = -sin θ Solve the equation for solutions in the interval [0, "). 7) sin x = 7) sec x = cos x 7) cos x = - cos x 75) sin x + sin x = 0 5
Solve the equation for solutions in the interval [0, 60 ). Round to the nearest degree. 76) cos θ = 1 77) tan θ = 5 6
Answer Key Testname: CH 5 & 6 EXAM REVIEW 1) - 5 ) 7 7 ) ) - 17 17 5) cos x 6) tan x 7) ± tan x + 1 8) - cot θ 9) sec θ 10) sin θ tan θ 11) sec θ csc θ 1) 0 1) 1 1) (1- sin x)(1 + sin x + sin x) 15) cos x 16) csc θ 17) tan θ 18) 1 19) tan θ csc θ = sin θ cos θ 1 sin θ = 1 cos θ = sec θ 0) (1 - cos x)(1 + cos x) = 1 - cos x = sin x 1) sec θ - 1 tan θ = sec θ - 1 tan θ sec θ + 1 sec θ + 1 = sec θ - 1 tan θ(sec θ + 1) = tan θ tan θ(sec θ + 1) = tan θ sec θ + 1 cos θ ) 1 - cot θ 1 + cot θ + 1 = 1 - cot θ csc θ + 1 = 1 csc θ - cot θ csc + 1 = sin θ - θ sin θ 1 sin θ + 1 = sin θ - cos θ + (sin θ + cos θ) = sin θ ) sin x cos x = sin x (1 - cos x) (cos x) = sin x (cos x - cos x). ) Identity 5) Not an identity 6) - 6-7) 8) -sin θ 9) - - 6 15 + 1 7
Answer Key Testname: CH 5 & 6 EXAM REVIEW 0) 186 + 15 18 1) csc 77 ) θ = 10 ) cot ) cos x + " 6 = cos x cos " 6 - sin x sin " 6 = cos x - 1 sin x. 5) cos (x - y) - cos (x + y) = cos x cos y + sin x sin y - ( cos x cos y - sin x sin y) = sin x sin y. 6) - 7) - 8) 9-8 5 9) tan x 0) sin 1) ) cos 8x 1 ) 81 ) - 119 169 " - θ = sin " cos θ - cos " 5) sin x cos x - sin x cos x sin θ = (-1) cos θ - 0 sin θ = - cos θ 6) cos(x) = cos(x + x) = cos(x) cos x - sin(x) sin x = (cos x - sin x) cos x - sin x cos x sin x = cos x - sin x cos x - sin x cos x = cos x - sin x cos x. 7) cos 10x + cos 6x 8) 5(cos 9 + cos 1 ) 9) - sin 1" sin -9" 50) - sin 1 sin(-1 ) 51) 1 + 5) - 1 + 5) 6 5) 10 + 1 55) sec u = 1 cos u = 1 + cos u = sec u sec u + 1 56) " 6 8
Answer Key Testname: CH 5 & 6 EXAM REVIEW 57) " 58) - " 59) " 60) 0 61).7958 6) 5 5 6) 6) 16 65 65) u + 1 u + 1 66) 5 u 67) 0, ", " 6, 5" 6 68) {0} 69) {70 } 70) {" + n"} 71) {0, 10, 180, 0 } " 7) 1, " 6, ", 7" 1, 7" 6, 1" 1, 5", 19" 1 7) {0} " 7) 8, 9" 8, 7" 8, 15" 8 75) 0, ", ", " 76) {0 } 77) {, 57, 1, 17, 1, 7, 0, 7 } 9