MATH 1040 CP 15 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 1) (sin x + cos x) 1 + sin x cos x =? 1) ) sec 4 x + sec x tan x - tan 4 x =? ) ) cos x + sin x cos x - sin x - cos x sin x =? ) 4) sec x csc x =? 4) 5) (1 + tan u)(1 - sin u) = 1 5) ) cot x + csc x = csc x - 1 ) Show that the equation is not an identity by finding a value of x for which both sides are defined but not equal. 7) sin x - sin x cosx = sin x 7) Rewrite the expression in terms of the given function or functions. 8) csc x + tan x csc x; cos x and sin x 8) Find the exact value of the expression. 9) cos 9-18 9) Identify and in the following expression which is the right side of the formula for cos ( - ). 10) cos 18 cos 9 + sin 18 sin 9 10) Find the exact value of the expression. 11) cos (15 ) cos (45 ) + sin (15 ) sin (45 ) 11) Write the expression as the cosine of an angle, knowing that the expression is the right side of the formula for cos ( - ) with particular values for and. 1) cos cos 9 18 + sin sin 1) 9 18 Use the given information to find the exact value of the expression. 1) sin = 4 5, lies in quadrant II, and cos =, lies in quadrant I Find cos ( - ). 1) 5 1
Find the exact value by using a sum or difference identity. 14) cos (15 + 0 ) 14) sin ( - ) 15) cos cos =? 15) Find the exact value of the expression. 1) sin 0 cos 40 + cos 0 sin 40 1) cos() 17) = cot - tan 17) cos sin 18) sin( - ) cos() = sin cos - sin cos 18) Use the given information to find the exact value of the expression. 19) sin = 5 1 15, lies in quadrant II, and cos =, lies in quadrant I Find sin ( - ). 19) 17 0) cos = - 7 5, lies in quadrant III, and sin = 1, lies in quadrant II Find cos ( + 5 ). 0) Find the exact value by using a difference identity. 1) tan 1 1) Use trigonometric identities to find the exact value. tan 5 + tan 85 ) 1 - tan 5 tan 85 ) ) tan x - 4 = tan x - 1 1 + tan x ) Find the exact value under the given conditions. 4) cos = - 4 5, < < ; sin = - 1 5, < < Find tan ( + ). 4) Rewrite the expression as a simplified expression containing one term. 5) sin -11 - cos -11 + + cos -11-11 - sin + 5)
Use the given information to find the exact value of the expression. ) cos = 5, lies in quadrant IV Find sin. ) 1 Use the figure to find the exact value of the trigonometric function. 7) Find tan. 7) 15 17 8 Write the expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. 8) tan 5 8 1 - tan 5 8 8) 9) cos 0 - sin 0 9) 0) sin x cos x + sin x cos x =? 0) 1) cos 4 = cos ( ) - 1 1) Rewrite the expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1. ) cos 4 x ) ) sin x ) Use a half-angle formula to find the exact value of the expression. 4) cos 5 1 4) 5) tan 15 5) Use the given information to find the exact value of the trigonometric function. ) cos = 1 4, csc > 0 Find sin. ) 7) sec = 4, lies in quadrant I Find cos. 7)
8) sec = - 5, lies in quadrant II Find sin. 8) 9) csc = - 5, tan > 0 Find cos. 9) 40) tan - cot =? 40) Use a graph in a [-,, ] by [-,, 1] viewing rectangle to complete the identity. 41) tan x 1 + tan x =? 41) Express the product as a sum or difference. 4) sin 4x sin 7x 4) 4) cos 11x cos x 4) 44) sin x cos 9x 44) Express the sum or difference as a product. 45) cos 5x x + cos 45) 4) sin 1x + sin 9x 4) 47) sin - sin - = tan cot sin + sin 47) sin x + sin 5x 48) cos x + cos 4x =? 48) Use substitution to determine whether the given x-value is a solution of the equation. 49) cos x + 1 = sin x, x = 5 4 49) 4
50) tan x =, x = -5 50) Find all solutions of the equation. 51) 9 cos x + = 7 cos x+ 5 51) 5) tan x sec x = - tan x 5) Solve the equation on the interval [0, ). 5) sin 4x = 5) 54) sin x = sin x 54) 55) sec x - = tan x 55) 5) sin x - cos x = 0 5) Solve the equation on the interval [0, ). 57) -tan x sin x = -tan x 57) 58) cos x + cos x sin x = 0 58) Solve the equation on the interval [0, ). 59) cos x + sin x - = 0 59) 0) sin x + 11 - sin x - 11 = 1 0) Use a calculator to solve the equation on the interval [0, ). Round the answer to two decimal places. 1) cos x = -0.8 1) Solve the problem. ) A coil of wire rotating in a magnetic field induces a voltage given by e = 0 sin ( t 4 - ), where t is time in seconds. Find the smallest positive time to produce a voltage of 10. ) 5
Answer Key Testname: MATH1040CP15 1) 1 ) sec 4 x - ) sec x csc x 4) sec x + csc x 5) (1 + tan u)(1 - sin u) = sec u cos u = 1 cos cos u = 1 u ) cot x + csc x = csc x - 1 + csc x = csc x - 1. 7) 4 8) 1 sin x cos x 9) 10) = 18, = 9 11) - 1 1) cos 1) - + 4 1 5 14) - ( + 1) 4 15) tan - tan 1) 17) cos() cos cos - sin sin cos cos = = cos sin cos sin cos sin - sin sin cos sin = cos sin 18) sin( - ) cos () = (sin cos - cos sin )(cos cos - sin sin ) = sin cos cos - sin cos sin - cos sin cos + cos sin sin - sin cos = sin cos (cos + sin ) - sin cos (sin + cos ) = sin cos - sin cos 19) 171 1 = cot - tan 0) 14 + 4 1 15 1) - ) - ) tan x - 4 = tan x - tan /4 1 + (tan x)(tan /4) = tan x - 1 1 + tan x. 4) - + 4 1 8 + 1
Answer Key Testname: MATH1040CP15 5) ) - 10 19 7) - 40 11 8) 1 9) 1 0) sin x 1) cos 4 = cos[( )] = cos ( ) - 1 + 4 cos x + cos 4x ) 8 ) 4 sin x - 1 4 sin x 4) 1-5) - ) 4 7) 10 4 8) 5 5 9) - 18-11 40) - cot 41) sin x 4) 1 (cos x - cos 11x) 4) 1 (cos 5x + cos x) 44) 1 (sin5x - sin 4x) 45) cos x cos x 4) sin 11x cos x 47) - sin cos sin - sin sin + sin = sin cos - = sin cos - - cos sin = tan - cot 48) sin 4x cos x 7
Answer Key Testname: MATH1040CP15 49) No 50) Yes 51) x = 4 + n or x = 4 + n 5) x = + n or x = + n or x = n 5) 1,,, 7 1, 7, 1 1, 5, 19 1 54) 0,,, 5 55) no solution 5) 4, 4, 5 4, 7 4 57) 0, 58),,, 11 59) 0,,, 5 0) 1).55,.7 ) seconds 8