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Math 8 CHAPTER 3 TEST FALL 2009 Find the inverse function f-1 of the function f. Find the domain of the function f and of its inverse function f-1. (Make the answers clear) 1) f(x) = -8 cos(2x) y = = -8 cos(2x) x = -8 cos(2y) x - = = cos(2y) 8 cos-1 y= x = 2y 8 1 x cos-1 2 8 1 x f-1 x = cos-1 2 8 -1 8 domain of f: all real numbers x x 8 1 -8 doamin of f-1 = -8, 8 Find the exact value of the expression. 2) csc-1 -2 = csc-1 -2 csc = - 2 1 sin = 2 = sin -1 =- 1 2 6 Use a calculator to find the value of the expression in radian measure rounded to two decimal places. 5 3) cot-1 2 5 cot-1 = 2 cot =- 5 2 tan =- 2 5 tan-1 - 2 = 5 -.38 Find the exact value of the expression. Do not use a calculator. 3 2 2 = tan-1 tan =4) tan-1 tan 5 5 5 1 Find the exact value, if any, of the composite function. If there is no value, say it is "not defined" and explain why it is not defined. Do not use a calculator. 5) sin(sin-1 1.3) undefined because sine is defined between -1 and 1 inclusive of both. 1.3 is out of this interval. Find the exact value of the expression. 2 6) sec sin -1 3 sin = - 2 3 -2 2 + x 2 = 3 2 x2 = 9 - 4 x=± 5 3 sec = 5 = 3 5 5 Given that f(x) = sin x, g(x) = cos x, and h(x) = tan x, find the exact value of the composite function. 5 1 = sin-1 cos = sin-1 = 7) f-1 g 3 3 2 6 Find the exact value of the expression. 5 5 8) cos tan -1 - csc-1 8 2 tan = 5 8 csc = 5 2 + 8 2 = r2 2 2 + x2 = 52 x2 = 25 - 4 25 + 64 = r 89 = r cos - 8 89 = 21 + 5 5 2 89 5 = 5 2 x = ± 21 8 21 10 8 21 + 10 + = 5 89 5 89 5 89 Find the exact value of the expression using a sum/difference formula AND a half angle formula. 3 2 1 1 2 3 2 1 6 2 6- 2 = sin = sin = = = there are other angle 9) sin 12 12 12 4 6 2 2 2 2 4 4 4 choices that would work sin 12 6 = sin 2 1 - cos = 2 6 3 2 1= 2 = 2- 3 2 2 = 2- 3 = 4 22 3 Use the information given about the angle , 0 2 , to find the exact value of the indicated trigonometric function. 3 3 3 < <2 Find sin . < < 10) sin = - , 5 2 2 4 2 1sin 2 = 2 4 5 = 5-4 5 2 = 1 10 = 10 10 Find the exact value of the expression. tan 65° - tan (-55°) = tan 65° - - 55° = tan 120° = 11) 1 + tan 65° tan (-55°) 2 3 Find the exact solution of the equation. 12) 7 cos-1 x - = 5 cos-1 x 2cos-1 x = cos-1 x = cos 2 =x 2 x=0 Find the real zeros of the trigonometric function on the interval 0 13) f(x) = 4 cos2 x - 3 x<2 0 = 4 cos2 x - 3 3 = cos2 x 4 3 = cos x 2 ± 5 1 , , 6 6 6 6 , CHOOSE 3 OUT OF THE FOUR ONLY. Solve the equation on the interval 0 2 = 14) cos 2 2 2 2 2 2 4 = = = = = 8 , = 2 = 8 4 2 - + 2k + 2k 4 8 + 2k 4 2 2 2k 2 = + 8k 8 = = + 8 = 11 8 = = 2 4 = + 8 8 <2 . = 8 4 + 2k + 2k 4 8 + 2k 4 + 2k 2 + 8k 8 k= 0 8 8 + 8 = 3 9 11 , , 8 8 8 3 17 8 = 8 + 16k 8 = 8 k= 1 15) 3 tan 2 3 tan 2 = tan - tan =0 tan 3 tan - 1 = 0 tan = 0 3 tan - 1 = 0 = 0, 3 tan = 1 tan = 0, , 6 1 3 = 3 3 = 6 , 7 6 5 6 , 16) cos(2 ) = sin 1 - 2sin 2 = sin 0 = 2sin 2 0 = 2sin 2sin 2sin -1=0 =1 1 = 2 sin = 17) sin + sin - 1 - 1 sin + 1 6 , sin sin = 6 6 + cos = - 2 1 sin 2 1 + cos 2 2 sin 2 + r= = - 2 cos 2 cos sin + sin cos sin + = -1 sin = 3 2 , 6 , 3 2 = - 2 - cos sin + +1=0 = -1 + 4 6 4 = 4 12 + 12 = 1+ 1= = -1 cos = 2 2 sin = = -1 = -1 3 2 - 4 = 2 2 2 4 4 2 2 = 4 Use a calculator to solve the equation on the interval 0 18) 2 sec = 5 5 sec = 2 cos = cos-1 2 5 2 = 5 66.42° 1.16rad 360 - 66.42° 293.58° 2 - 1.16 5.12 CHOOSE 3 OUT OF THE 5 ONLY. Establish the identity. sin 3 - cos3 19) sin - cos sin - cos sin 2 + sin cos sin - cos sin2 + sin cos 1 + sin cos 20) sin cos - cos sin sin sin = cos sin - = cot - cot cos2 = cot x x 2 - sin 2 2 x x x x - 2cos sin + sin2 2 2 2 2 1 - sin 2 1 - sin x x 2 + cos2 cos = = = 1 + sin = cot cos sin = 1 + sin + cos2 sin( - ) sin sin 21) cos < 2 . Round the answer to two decimal places. cos - cot - cot = 1 - sin x = = = 1 - sin x 5 22) sin(7 ) + sin(3 ) 2 sin(5 ) 2sin 7 +3 2 cos = cos(2 ) 7 -3 2 = 2 sin(5 ) 2sin 10 2 cos 4 2 = 2 sin(5 ) 2sin cos 2 sin(5 ) = cos(2 ) = cos(2 ) 23) ln 1 + sin u + ln 1 - sin u ln 1 + sin u 1 - sin u ln 1 - sin2 u ln cos 2u 2 ln cos u = 2 ln cos u = = = = 2 ln cos u BONUS. 10 POINTS CHOOSE ONLY 1 OF THE PROBLEMS BELOW Given that f(x) = sin x, g(x) = cos x, and h(x) = cot x, evaluate the given function. The point (x, 3), on the circle 1 x2 + y2 = 4, also lies on the terminal side of an angle in standard position. The point , y , on the circle x 2 + y2 = 1, 4 also lies on the terminal side of an angle 24) h( - ) in quadrant IV. Find the exact value of the expression. 12 25) cos 3 tan-1 5 6