Math 370 Exam 3 Review Name
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1 Math 370 Exam 3 Review Name The following problems will give you an idea of the concepts covered on the exam. Note that the review questions may not be formatted like those on the exam. You should complete all problems to the best of your ability and be ready to discuss any questions you have during the review session. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Complete the identity. 1) sin 3x sin 5x cos 3x cos 5x =? 1) 2) 3) A) cos2 2x - cos 2 8x C) sin2 30x A B) cos 2 30x D) cos2 8x + cos 2 2x sin ( + ) sin ( - ) =? 2) A) tan - tan tan + tan C B) tan ( + ) tan - tan C) tan + tan tan - tan D) tan - tan tan ( + ) cos ( - ) cos cos =? 3) A) 1 + cot tan B) tan tan + cot C) 1 + tan tan D) 1 + cot cot C ) sin 2 x + tan 2 x + cos 2 x =? ) A) sin x B) cot 3 x C) tan 2 x D) sec 2 x D 5) 2 tan x - (1 + tan x) 2 =? 5) A) - sec 2 x B) 1 C) 1 - sin x D) 0 A 6) sin x cos x + cos x sin x =? 6) A) 1 + cot x B) -2 tan 2 x C) sec x csc x D) sin x tan x C
2 Describe the graph using another equation. 7) y = cos x cos x - 2 7) A) -2cos x B) -sin x C) -2sin x D) -cos x C Show that the equation is not an identity by finding a value of x for which both sides are defined but not equal. 8) sin 5x + sin 7x = sin 11x 8) A) 0 B) C) 2 D) 2 D The graph of a polar equation is given. Select the polar equation for the graph. 9) 9) A) r sin = 2 B) r = cos C) r = sin D) r = 2 C SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Convert the polar equation to a rectangular equation. 10) r 2 sin 2 = 10) xy = 2 11) r = 8 cos + 9 sin 11) x 2 + y 2 = 8x + 9y
3 Use the dot product to determine whether the vectors are parallel, orthogonal, or neither. 12) v = 2i + j, w = i - 2j 12) Verify the identity. orthogonal 13) tan 2 x (1 + cos 2x) = 1 - cos 2x 13) tan 2 x (1 + cos 2x) = 1 - cos 2x (1 + cos 2x) = 1 - cos 2x 1 + cos 2x 1) (1 + tan 2 u)(1 - sin 2 u) = 1 1) (1 + tan 2 u)(1 - sin 2 u) = sec 2 u cos 2 1 u = cos 2 cos 2 u = 1 u Find a. If necessary, round your answer to the nearest hundredth. 15) 15) Find all solutions of the equation. 16) tan x sec x = -2 tan x 16) x = n or x = 3 + 2n or x = n 17) 5 sin x = 3 sin x ) x = + 2n or x = 3 + 2n Find all the complex roots. Write the answer in the indicated form. 18) The complex square roots of + 3i (rectangular form) 18) 6 + 2i, - 6-2i 19) The complex fourth roots of -16 (rectangular form) 19) 2 + 2i, 2-2i, i, - 2-2i Find the absolute value of the complex number. 20) z = i 20) 2 7
4 Find the angle between the given vectors. Round to the nearest tenth of a degree. 21) u = -2i + 5j, v = i - 6j 21) Find the exact value by using a difference identity. 22) tan ) 2-3 Find the exact value by using a sum or difference identity. 23) sin ) 2( 3-1) Find the exact value of the expression. (Do not use a calculator.) 2) cos (155 ) cos (35 ) + sin (155 ) sin (35 ) 2) Find the product of the complex numbers. Leave answer in polar form. 25) z1 = i z2 = i 25) 2 2 cos 5 + i sin 5 Find projwv. 26) v = -3i - 3j; w = 3i + j 26) - 6 (3i + j) 5 Find the quotient z 1 of the complex numbers. Leave answer in polar form. z2 27) z1 = 3 cos 7 + i sin 7 27) z2 = 6 cos 9 + i sin cos i sin 3 2 Find the specified vector or scalar. 28) u = -i + 1j and v = i + 1j; Find u + v. 28) 2 Find the unit vector that has the same direction as the vector v. 29) v = 5i - 12j 29) u = 5 13 i j
5 Graph the polar equation. 30) r cos = -2 30)
6 31) r = sin 2 31) Polar coordinates of a point are given. Find the rectangular coordinates of the point. 32) (3, -135 ) 32) -3 2, Rewrite the expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1. 33) 2 sin 2 x cos 2 x 33) 1-1 cos x 3) sin 3 x 3) 3 sin x - 1 sin 3x
7 Sketch the vector as a position vector and find its magnitude. 35) v = -i + j 35) v = 2 Solve the equation in the complex number system. 36) x 5-1 = 0 36) 1, cos(72 )+ i sin(72 ), cos(1 ) + i sin(1 ), cos(216 ) + i sin(216 ), cos(288 ) + i sin(288 ) Solve the equation on the interval [0, 2 ). 37) tan 2 x sin x = tan 2 x 37) 0, Solve the equation on the interval [0, 2 ). 38) cos x + - cos x - = 1 38), 39) sin 2 x - cos 2 x = 0 39), 3, 5, 7
8 0) sin x = , 6, 2 3, 7 12, 7 6, 13 12, 5 3, ) Solve the problem. 1) A force of 6 pounds acts in the direction of 6 to the horizontal. The force moves an object along a straight line from the point (6, 8) to the point (15, 12), with distance measured in feet. Find the work done by the force. Round the answer to one decimal place, if necessary ft-lb 2) A power boat in still water maintains a speed of 35 miles per hour. The boat heads directly across a river perpendicular to the current which has a speed of 8 miles per hour. Find the actual speed and direction of the boat. 36 miles per hour; 13 off course 3) One rope pulls a barge directly east with a force of 79 newtons, and another rope pulls the barge directly north with a force of 87 newtons. Find the magnitude of the resultant force acting on the barge. 118 newtons ) A painter needs to cover a triangular region 60 meters by 68 meters by 71 meters. A can of paint covers 70 square meters. How many cans will be needed? 27 cans 5) Two points A and B are on opposite sides of a building. A surveyor selects a third point C to place a transit. Point C is 51 feet from point A and 72 feet from point B. The angle ACB is 58. How far apart are points A and B? 62. feet 6) A generator produces an alternating current according to the equation I = 8 sin 122 t, where t is time in seconds and I is the current in amperes. What is the smallest time t such that I = 2? second 1) 2) 3) ) 5) 6) Test the equation for symmetry with respect to the given axis, line, or pole. 7) r = 2 sin 3 ; the line = 2 7) has symmetry with respect to the line = 2 Two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round lengths to the nearest tenth and angle measures to the nearest degree. 8) B = 13, b =.3, a = ) A1 = 19, C1 = 18, c1 = 10.1; A2 = 161, C2 = 6, c2 = 2
9 9) B = 88, b = 2, a = 23 9) no triangle Use DeMoivre's Theorem to find the indicated power of the complex number. Write the answer in rectangular form. 50) (- 3 + i)6 50) -6 Use a half-angle formula to find the exact value of the expression. 51) cos ) ) sin 75 52) Use the figure to find the exact value of the trigonometric function. 53) Find cos 2. 53) Use the given information to find the exact value of the trigonometric function. 5) sin = 1, tan > 0 Find cos 2. 5) Use the given information to find the exact value of the expression. 55) sin = - 3 5, lies in quadrant IV, and cos = - 21, lies in quadrant III Find sin ( - 5 ). 55) Use the given vectors to find the specified scalar. 56) u = -5i + 3j, v = 5i - 6j, w = -3i + 12j; Find u w + v w. 56) -36
10 Use the graph to complete the identity. sin 2x + sin 6x 57) cos 6x - cos 2x =? 57) -cot 2x Use the result e i = cos + i sin to plot the complex number. 58) 5e ( i)/2 58)
11 Use the vectors v, u, w, and z to draw the indicated vector. 59) u 59) 60) u + z 60)
12 61) v - w 61) Write the complex number in polar form. Express the argument in degrees. 62) i 62) 10(cos i sin ) Write the complex number in rectangular form. 63) 7(cos i sin 120 ) 63) i Write the vector v in terms of i and j whose magnitude v and direction angle are given. 6) v = 8, = 30 6) v = 3i + j
Math 370 Exam 3 Review Name
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