PRINTABLE VERSION Practice Final Question 1 Suppose that is an acute angle of a right triangle and that sec() = a) csc() = 33 7 b) csc() = 7 33 33 c) 33 4 csc() = 33 d) csc() = 33 4 e) csc() = 7 33 33 f) None of the above. tan() = 7 . Find csc() and tan() . 4 33 4 tan() = 4 33 33 33 7 tan() = 33 tan() = 33 7 tan() = 33 4 Question 2 In the figure below, B is a right angle, mBAC = 30 , AC = 6, and CD = 5 . Find AD. Note: The triangle may not be drawn to scale. a) 2 91 b) 91 c) 91 d) 182 e) 6 f) None of the above. Question 3 Find the area of the sector of a circle with central angle = 240 and radius r = 9 cm. a) 24 cm2 b) 12 cm2 c) 6 cm2 d) 54 cm2 e) 108 cm2 f) None of the above. Question 4 7 11 + cos + tan( ) . ( 6 ) ( 6 ) 6 Evaluate: sin a) 3 + 5 3 6 b) 3 3 6 c) 3 3 6 d) 7 3 + 3 6 e) 3 + 5 3 6 f) None of the above. Question 5 Evaluate: sin(210 ) + cos(315 ). a) 3 + 2 2 b) 1 + 2 2 c) 3 + 2 2 d) 2 1 2 e) 2 1 2 f) None of the above. Question 6 Evaluate: sin(45 ) + cos(225 ) a) 2 4 b) 2 c) 0 d) 2 + 2 2 e) 2 2 f) None of the above. Question 7 Evaluate: cos( ) + sin . ( 3 ) 3 a) 0 b) 1 + 3 2 2 1 + 3 2 c) d) 5 2 e) 3 f) None of the above. Question 8 Let P(x, y) denote the point where the terminal side of an angle meets the unit circle. If P is in Quadrant IV and x = 3 , find tan() . 7 a) 40 9 b) 2 10 3 c) d) 10 3 20 e) 10 3 20 f) None of the above. 2 10 3 Question 9 Which of the following is equivalent to the expression below? tan() + a) 1 + sin2 () b) sec() c) sin() d) 1 cos() 1 + sin() e) csc() f) None of the above. Question 10 Which of the following is equivalent to the expression below? cos() 1 + sin() + 1 + sin() cos() a) 1 b) 2 csc() c) 2 sin() d) 2 e) 2 sec() f) None of the above. Question 11 Which of these is an equation of one of the asymptotes of the function f (x) = 5 csc 1 x + 2? (3 ) 3 4 a) x= b) x = 3 c) x= d) x=2 e) x= f) None of the above. 3 2 Question 12 Evaluate the following expression. Do not use a calculator. If undefined, state Undefined. 2 cos1 + tan1 (3 ) ( 2 ) 3 a) b) 12 c) 13 12 d) Undefined e) f) None of the above. 5 12 Question 13 Evaluate the following expression. Do not use a calculator. If undefined, state Undefined. 3 cos arcsin ( ( 5 )) 4 5 a) b) 3 5 c) 4 5 d) e) Undefined f) None of the above. 3 5 Question 14 Given cos(x) = 15 3 with 0 < x < 90 and cos(y) = with 270 < y < 360 find cos(x + y) . 17 5 a) 84 85 b) 77 85 c) d) 13 85 e) 126 85 f) None of the above. 36 85 Question 15 Given sin(x) = 4 7 with 0 < x < 90 , and sin(y) = with 0 < y < 90 . Find sin(x + y) . 9 10 a) 15 16 3 + 7 90 b) + 7 65 4 51 90 c) 4 3 + 7 5 \\displaystyle \\frac{4\\,\\sqrt {3}+7\\,\\sqrt {5}}{90} 90 d) + 7 65 4 51 \\displaystyle -\\frac{-4\\,\\sqrt {51}+7\\,\\sqrt {65}}{90} 90 e) \\displaystyle \\frac{659}{90} f) None of the above. 659 90 Question 16 Given sin(x)=\\displaystyle -\\frac{6}{7}sin(x) = a) \\displaystyle \\frac{-8\\sqrt{3}}{7} 83 7 6 with 180 < x < 270. Find sin(2x)sin(2x) . 7 1213 49 b) \\displaystyle \\frac{12\\sqrt{13}}{49} c) 213 \\displaystyle \\frac{2\\sqrt{13}}{49} 49 d) 13 2 \\displaystyle \\frac{-2\\sqrt{13}}{49} 49 e) \\displaystyle \\frac{-12\\sqrt{13}}{49} f) None of the above. 1213 49 Question 17 Solve \\,\\displaystyle \\sin\\left(\\frac{1}{3}x+\\displaystyle \\frac{\\pi}{4}\ ight)=\\displaystyle - 1 3 \\frac{\\sqrt{3}}{2}\\, sin x+ = over the interval \\,\\displaystyle \\left[\\displaystyle (3 4) 2 3 21 \\frac{3\\,\\pi}{4},\\displaystyle \\frac{21\\,\\pi}{4}\ ight) , . [ 4 4 ) a) \\displaystyle x=\\displaystyle \\frac{13\\,\\pi}{4},\\, x=\\displaystyle \\frac{17\\,\\pi}{4} x= 13 17 ,x= 4 4 b) \\displaystyle x=\\displaystyle \\frac{13\\,\\pi}{4}x = 13 4 c) \\displaystyle x=\\displaystyle \\frac{17\\,\\pi}{4}x = 17 4 d) \\displaystyle x=\\displaystyle \\frac{11\\,\\pi}{4},\\, x=\\displaystyle \\frac{19\\,\\pi}{4} x= 11 19 ,x= 4 4 e) \\displaystyle x=\\displaystyle \\frac{11\\,\\pi}{4}x = f) None of the above. 11 4 Question 18 2 Solve \\,\\displaystyle 5\\cos^2(2 x)-\\displaystyle \\frac{5}{2}=0\\, 5 cos (2x) \\,\\displaystyle \\left[0,\\pi\ ight) [0, ) . 5 = 0 over the interval 2 8 a) \\displaystyle x=\\displaystyle \\frac{\\pi}{8}x = b) \\displaystyle x=\\displaystyle \\frac{\\pi}{8},\\, x=\\displaystyle \\frac{3\\,\\pi}{8},\\, x=\\displaystyle \\frac{5\\,\\pi}{8},\\, x=\\displaystyle \\frac{7\\,\\pi}{8}x = 3 5 7 ,x= ,x= ,x= 8 8 8 8 3 5 ,x= 8 8 c) \\displaystyle x=\\displaystyle \\frac{3\\,\\pi}{8},\\, x=\\displaystyle \\frac{5\\,\\pi}{8}x = d) \\displaystyle x=\\displaystyle \\frac{\\pi}{8},\\, x=\\displaystyle \\frac{3\\,\\pi}{8}x = 3 ,x= 8 8 e) \\displaystyle x=\\displaystyle \\frac{\\pi}{8},\\, x=\\displaystyle \\frac{7\\,\\pi}{8}x = 7 ,x= 8 8 f) None of the above. Question 19 Find the phase shift for the following function: \\displaystyle f(x)=5sin\\left(\\displaystyle \\frac{\\pi}{5}x+\\displaystyle \\pi\ ight)+4 f (x) = 5sin( x + ) + 4 5 a) 4 left b) right c) left d) 5 right e) 5 left f) None of the above. Question 20 Identify the equation whose graph is shown below. The point \\left(\\displaystyle \\pi,\\displaystyle \\frac{1} ( {2}\ ight) , 1 is on the graph. 2) a) \\displaystyle y=cos\\left(x\ ight)y = cos(x) b) \\displaystyle y=\\displaystyle -\\frac{1}{2}\\,cos\\left(x\ ight)y = 1 cos(x) 2 c) \\displaystyle y=\\displaystyle -\\frac{1}{2}\\,sin\\left(x\ ight)y = 1 sin(x) 2 d) \\displaystyle y=\\displaystyle \\frac{1}{2}\\,cos\\left(x\ ight)y = e) \\displaystyle y=\\displaystyle \\frac{1}{2}\\,cos\\left(2x\ ight)y = f) None of the above. 1 cos(x) 2 1 cos(2x) 2 Question 21 Which of these is an equation of one of the asymptotes of the following function? \\displaystyle f(x)=5\\,sec\\left(\\displaystyle \\frac{1}{2}\\pi x-\\pi\ ight)f (x) = 5 sec a) \\displaystyle x=4x = 4 1 x (2 ) b) \\displaystyle x=\\displaystyle \\frac{3\\pi}{4}x = c) \\displaystyle x=3x = 3 d) \\displaystyle x=\\displaystyle 3\\pix = 3 e) \\displaystyle x=\\displaystyle \\frac{3}{2}x = f) None of the above. 3 4 3 2 Question 22 Given \\displaystyle \\sec \\left( x \ ight) =-5sec(x) = 5 with 90^{\\circ} < x < 180^{\\circ} 90 < x < 180 . Find \\displaystyle \\cos \\left( 2\\,x \ ight) cos(2 x) . 24 25 a) \\displaystyle -\\frac{24}{25} b) \\displaystyle \\frac{-4\\sqrt{6}}{25} c) \\displaystyle -\\frac{23}{25} d) \\displaystyle \\frac{23}{25} e) 26 \\displaystyle \\frac{2\\sqrt{6}}{25} 25 f) None of the above. 46 25 23 25 23 25 Question 23 Give the number of solutions to the following equation on the interval \\left[\\,0, 2\\pi\\,\ ight)[ 0, 2 ). 2 \\displaystyle cos^2(x)=\\displaystyle \\frac{1}{4}cos (x) = a) 3 b) 0 c) 1 d) 4 1 4 e) 2 f) None of the above. Question 24 A string running from the ground to the top of a fence has an angle of elevation of 45. The string is 6 feet long. What is the distance between the fence and where the string is pegged to the ground? a) 6 b) 33 c) 32 d) 3 e) 12 f) None of the above. Question 25 Suppose that you are headed toward a plateau 50 meters high. If the angle of elevation to the top of the plateau is 27, how far are you from the base of the plateau, in meters? a) \\displaystyle 50tan(27^{\\circ})50tan(27 ) b) \\displaystyle 50sin(27^{\\circ})50sin(27 ) c) \\displaystyle 50cos(27^{\\circ})50cos(27 ) d) \\displaystyle \\frac{50}{tan(27^{\\circ})} 50 tan(27 ) e) \\displaystyle \\frac{50}{sin(27^{\\circ})} 50 sin(27 ) f) None of the above. Question 26 In right triangle ABCABC with m\\angle C = 90^{\\circ}mC = 90 , m\\angle B =28^{\\circ}, mB = 28 , and AB=7AB = 7 . Find ACAC . a) \\displaystyle \\frac{7}{2} b) 7\\,tan(28^{\\circ}) c) 7\\,sin(28^{\\circ}) d) sin(28^{\\circ}) e) 7\\,cos(28^{\\circ}) f) None of the above. Question 27 Find the area of triangle \\,GHJ\\, if \\angle G = 120^{\\circ}, h = 6 and j = 17. a) \\displaystyle 51\\sqrt{3} b) \\displaystyle \\frac{51\\sqrt{3}}{2} c) \\displaystyle \\frac{51}{2} d) \\displaystyle 9\\sqrt{3} e) \\displaystyle \\frac{289}{2} f) None of the above. Question 28 Given triangle ABC, m\\angle A = 45^{\\circ}, \\displaystyle {\\it AB}=6 and AC=\\displaystyle \\sqrt{2}\\,. Find BC. a) 8 b) \\displaystyle \\sqrt{26} c) 3 d) \\displaystyle \\sqrt{38} e) \\sqrt{10-\\displaystyle \\sqrt{2}} f) None of the above. Question 29 ABC\\, is a triangle with \\angle A = 60^{\\circ}, \\angle B = 45^{\\circ}, and \\displaystyle {\\it BC}=8. Find \\,AC. a) 64 b) \\displaystyle \\frac{16}{3} c) \\displaystyle \\frac{8\\sqrt{6}}{3} d) \\displaystyle 4\\sqrt{6} e) 12 f) None of the above. Question 30 ABC\\, is a triangle with \\,m\\angle A = 60^{\\circ}, BC = \\displaystyle 7\\sqrt{3} and AC=\\displaystyle 7\\sqrt{2}\\,. Find all possible measures for \\angle B. a) 60 b) 120 c) 45 or 135 d) 45 e) 30 or 150 f) None of the above. Question 31 State the coordinates of the vertex for the given parabola. {y}^{2}-2\\,x+12\\,y+26=0 a) \\left(-5,-3\ ight) b) \\left(5,6\ ight) c) \\left(-6,-5\ ight) d) \\left(-5,-6\ ight) e) \\left(6,5\ ight) f) None of the above. Question 32 Find the coordinates of the center for the given circle. \\displaystyle 10\\,{x}^{2}+10\\,{y}^{2}+240\\,x-180\\,y=0 a) \\left(-24,18\ ight) b) \\left(24,-18\ ight) c) \\left(12,-9\ ight) d) \\left(-12,9\ ight) e) \\left(-120,90\ ight) f) None of the above. Question 33 State the coordinates of the foci for the given ellipse. \\displaystyle \\displaystyle \\frac{{x}^{2}}{36}+\\displaystyle \\frac{{y}^{2}}{49}=1 a) \\left(\\displaystyle \\sqrt{85},0\ ight) and \\left(\\displaystyle -\\sqrt{85},0\ ight) b) \\left(0,\\displaystyle -\\sqrt{13}\ ight) and \\left(0,\\displaystyle \\sqrt{13}\ ight) c) \\left(0,\\displaystyle -\\sqrt{85}\ ight) and \\left(0,\\displaystyle \\sqrt{85}\ ight) d) \\left(\\displaystyle \\sqrt{85},\\displaystyle -\\sqrt{13}\ ight) and \\left(\\displaystyle \\sqrt{85},\\displaystyle \\sqrt{13}\ ight) e) \\left(\\displaystyle -\\sqrt{13},0\ ight) and \\left(\\displaystyle \\sqrt{13},0\ ight) f) None of the above. Question 34 Write the following in standard form for an ellipse.9\\,{x}^{2}+25\\,{y}^{2}-54\\,x+150\\,y+81=0 a) \\displaystyle \\displaystyle \\frac{ \\left( x-3 \ ight) ^{2}}{9}+\\displaystyle \\frac{ \\left( y+3 \ ight) ^{2}}{25}=1 b) \\displaystyle \\displaystyle \\frac{ \\left( x-3 \ ight) ^{2}}{25}+\\displaystyle \\frac{ \\left( y+3 \ ight) ^{2}}{9}=1 c) \\displaystyle \\displaystyle \\frac{ \\left( x-3 \ ight) ^{2}}{15}+\\displaystyle \\frac{ \\left( y+3 \ ight) ^{2}}{15}=1 d) \\displaystyle \\displaystyle \\frac{ \\left( x+3 \ ight) ^{2}}{25}+\\displaystyle \\frac{ \\left( y-3 \ ight) ^{2}}{9}=1 e) \\displaystyle \\displaystyle \\frac{ \\left( x+3 \ ight) ^{2}}{9}+\\displaystyle \\frac{ \\left( y-3 \ ight) ^{2}}{25}=1 f) None of the above. Question 35 State the equations of the asymptotes for the following: \\displaystyle \\displaystyle \\frac{{x}^{2}}{16}-\\displaystyle \\frac{{y}^{2}}{25}=1 a) \\displaystyle y=\\displaystyle \\frac{4}{5}x , y=\\displaystyle -\\frac{4}{5}x b) \\displaystyle x=0 , y=0 c) \\displaystyle y=5x , y=-5x d) \\displaystyle y=\\displaystyle \\frac{5}{4}x , y=\\displaystyle -\\frac{5}{4}x e) \\displaystyle y=4x , y=-4x f) None of the above. Question 36 Find the point(s) of intersection for the following functions:x={y}^{2}-10\\,y x-2\\,y=-35 a) \\left(-16,8\ ight) and \\left(-24,6\ ight) b) \\left(-21,7\ ight) and \\left(-25,5\ ight) c) \\left(-24,4\ ight) and \\left(-16,2\ ight) d) \\left(-24,6\ ight) and \\left(-24,4\ ight) e) There are no points of intersection. f) None of the above. Question 37 Find the sum of the vectors \\,\\vec{u}=-6\\vec{i}+2\\vec{j}\\, and \\,\\vec{v}=-2\\vec{i}-5\\vec{j}\\,. a) \\displaystyle \\vec{v}=-7\\vec{i}-4\\vec{j} b) \\displaystyle \\vec{v}=-8\\vec{i}-3\\vec{j} c) \\displaystyle \\vec{v}=-8\\vec{i}-8\\vec{j} d) \\displaystyle \\vec{v}=-3\\vec{i}-8\\vec{j} e) \\displaystyle \\vec{v}=-4\\vec{i}-7\\vec{j} f) None of the above. Question 38 Given vectors \\,\\vec{u}=\\langle-6,-3\ angle\\, and \\,\\vec{v}=\\langle-3,5\ angle, find \\, -3\\vec{u}+4\\vec{v}. a) \\displaystyle \\vec{v}=\\langle-27,-15\ angle b) \\displaystyle \\vec{v}=\\langle27,8\ angle c) \\displaystyle \\vec{v}=\\langle6,29\ angle d) \\displaystyle \\vec{v}=\\langle29,6\ angle e) \\displaystyle \\vec{v}=\\langle-15,-27\ angle f) None of the above. Question 39 Given the vector \\,\\vec{v}= \\bigg\\langle3,\\displaystyle 3\\sqrt{3}\\bigg\ angle, find the direction angle of this vector. a) \\displaystyle 0 b) \\displaystyle \\displaystyle \\frac{5\\pi}{6} c) \\displaystyle \\displaystyle -\\frac{\\pi}{3} d) \\displaystyle \\displaystyle \\frac{\\pi}{3} e) \\displaystyle \\displaystyle \\frac{\\pi}{6} f) None of the above. Question 40 Find the vector \\,\\vec{v}\\, given that its magnitude is \\, 12\\, and it makes an angle \\,\\theta=\\displaystyle \\frac{5\\pi}{6}\\, with the positive \\, x-axis. a) \\displaystyle \\vec{v}=\\displaystyle -6\\sqrt{3}\\vec{i}-6\\vec{j} b) \\displaystyle \\vec{v}=6\\vec{i}-\\displaystyle 6\\sqrt{3}\\vec{j} c) \\displaystyle \\vec{v}=\\displaystyle 6\\sqrt{3}\\vec{i}-6\\vec{j} d) \\displaystyle \\vec{v}=\\displaystyle 6\\sqrt{3}\\vec{i}+6\\vec{j} e) \\displaystyle \\vec{v}=\\displaystyle -6\\sqrt{3}\\vec{i}+6\\vec{j} f) None of the above. Question 41 Given vector \\,\\vec{v}=5\\vec{i}+10\\vec{j}\\,, which of the following is a vector that is orthogonal to \\,\\vec{v}? a) \\displaystyle -5\\vec{i}-10\\vec{j} b) \\displaystyle -10\\vec{i}-5\\vec{j} c) \\displaystyle -20\\vec{i}+10\\vec{j} d) \\displaystyle -5\\vec{i}+10\\vec{j} e) \\displaystyle 20\\vec{i}+5\\vec{j} f) None of the above. Question 42 Give all possible polar coordinates for the point \\left(-1,\\displaystyle \\sqrt{3}\ ight) given in rectangular coordinates. (In the choices below, n represents any integer.) a) \\left[4,\\displaystyle \\frac{2\\pi}{3}+2n\\pi\ ight], \\left[-4,\\displaystyle \\frac{5\\pi}{3}+2n\\pi\ ight] b) \\left[2,\\displaystyle \\frac{2\\pi}{3}+2n\\pi\ ight], \\left[-2,\\displaystyle \\frac{5\\pi}{3}+2n\\pi\ ight] c) \\left[-2,\\displaystyle \\frac{2\\pi}{3}+2n\\pi\ ight], \\left[2,\\displaystyle \\frac{5\\pi}{3}+2n\\pi\ ight] d) \\left[1,\\displaystyle -\\frac{2\\pi}{3}+2n\\pi\ ight], \\left[-1,\\displaystyle -\\frac{5\\pi}{3}+2n\\pi\ ight] e) \\left[2,\\displaystyle \\frac{5\\pi}{3}+2n\\pi\ ight], \\left[-2,\\displaystyle \\frac{2\\pi}{3}+2n\\pi\ ight] f) None of the above. Question 43 Write the equation \\; \\left( {x}^{2}+{y}^{2} \ ight) ^{2}=6\\,xy\\; in polar coordinates. a) \\displaystyle {r}^{2}=6\\,\\sin \\left( 2\\,\\theta \ ight) b) \\displaystyle {r}^{2}=6\\,\\cos \\left( 2\\,\\theta \ ight) c) \\displaystyle {r}^{2}=6\\,\\sin \\left( \\theta \ ight) \\cos \\left( \\theta \ ight) d) \\displaystyle r=6\\,\\cos \\left( 2\\,\\theta \ ight) e) \\displaystyle {r}^{2}=1/6\\,\\sin \\left( \\theta \ ight) \\cos \\left( \\theta \ ight) f) None of the above. Question 44 Which of the following shows the correct sketch of the polar curve \\;r=\\sin \\left( \\theta \ ight) ? a) b) c) d) e) f) None of the above. Question 45 Convert the equation \\displaystyle 5\\,{x}^{2}+5\\,{y}^{2}-60\\,x=0 into polar form. a) \\displaystyle r=6\\,\\sin \\left( \\theta \ ight) b) \\displaystyle r=12\\,\\cos \\left( \\theta \ ight) c) \\displaystyle r=12\\,\\sin \\left( \\theta \ ight) d) \\displaystyle r=12 e) \\displaystyle r=6\\,\\cos \\left( \\theta \ ight) f) None of the above. Question 46 Let \\displaystyle f(x)=5e^{3x} and \\displaystyle g(x)=\\frac{2}{3}\\,\\ln(5x). Find \\displaystyle (f\\circ g) (2). a) \\displaystyle75 b) \\displaystyle20 c) \\displaystyle50 d) \\displaystyle125 e) \\displaystyle500 f) None of the above. Question 47 Let \\;f(x)=\\log_{2}(x+1)\\; and \\;g(x)=4\\cdot 2^x. Find the value of \\,f^{-1}(2)+(g\\circ f)(2). a) \\displaystyle16 b) \\displaystyle11 c) \\displaystyle17 d) \\displaystyle9 e) \\displaystyle15 f) None of the above