2. Use fundamental identities and/or the complementary angle theorem to find the value of the following expression: tan 55. (e) BC?

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1 Math Final Exam. In the given right triangle a = 7 and B = 40. Find the hypotenuse c. (a) 7 sec 50 (b) 7 cos 40 (c) 7 csc 40 (d) 7 sin 50 (e) 7 sec 40 (f) 7 sin Use fundamental identities and/or the complementary angle theorem to find the value of the following expression: tan c 40 a = 7 b (a) 0 (b) 8 8 (c) 2 (d) (e) 5 8 (f) 3 3. The angle of elevation of the sun is 35 at the instant the shadow cast by a tower is 744 feet long. What is the height of the tower in feet? (a) 744 sin(35 ) ft. (b) 744 tan(35 ) ft. (c)744 cos(35 ) ft. (d) 4>?(56 ) ft. (e) BC?(56 ) ft. (f) 234(56 ) ft. 4. Mike is trying to make a triangle with sides a, b, and c so that a = 9, b = 0, and A = 30. How many triangles does the given information produce? (assume that in this triangle, side a opposite of angle A, side b is opposite of angle B, and side c is opposite of angle C) (a) No triangles (b) One triangle (c) Two triangles (d) None of the above (e) Not enough information to tell 5. Given the following triangle, find the measure of the missing side length x. Simplify where possible. (a) 2 3 sin 0 (b) 2 3 sin 0 (c) 6 sin 0 (d) 2 3 cos 0 (e) 8 sin 0 (f) 8 cos Given the following triangle, find the measure of the missing side length x. Simplify where possible. x #5 (a) A 4>? 86 4>? JKK (b) A 4>? 86 4>? 66 (c) A 4>? 66 4>? #6 (d) 4 sin L JJ A 4>? JKK M (e) 8K 4>? 66 (f) A 4>? 66 4>? JKK x 25

2 7. Given the following triangle, use the law of cosines to find the measure of the missing side length x. Simplify where possible #7 (a) N (b) N (c) 9 (d) N (e) N (f) (Drawing not to scale) 5 x 8. Sally s father just built her a new slide. The actual slide section is 6 feet long and the ladder up to the top of the slide is 4 feet long. If the distance between the base of the ladder and the base of the slide section is 5 feet, what is the measure of the angle that the slide makes with the ground (the measure of θ in the picture to the right). Keep in mind that the ladder does not necessarily make a 90 angle with the ground and that the drawing is not to scale. (a) cos QJ 5 8 (b) cos QJ 6 R (c) cos QJ 5 A (d) cos QJ S JR 4 ft e) none of the above θ 5 ft 6 ft 9. Find the area of the given triangle. (a) 5 cos 20 in 8 (b) 0 cos 20 in 8 (c) 5 in 8 (d) 5 sin 20 in 8 (e) 0 sin 20 in 8 (f) none of the above 2 in in. #9 0. A dog is enclosed in a fenced area that is triangular with side lengths 4 yards, 6 yards, and 8 yards. What is the area of this enclosed region? (a) 5 yd 8 (b) 3 5 yd 8 (c) 9 yd 8 (d) 2 yd 8 (e) 2 20 yd 8 (f) none of the above. Billy is playing on his swing. One full swing (front to back to front) takes 8 seconds and at the peak of his swing he is at an angle of 30 with the vertical. If the length of his swing (the length of the rope from where it is attached to where he sits) is 6 feet long, and we ignore all resistive forces, write an equation that relates his horizontal displacement (from the rest position) after time t (in seconds). (start with Billy being at the peak of his swing at time t=0). [Hint: the horizontal displacement would follow simple harmonic motion] (a) d(t) = 3 cos L \ tm (b) d(t) = 6 cos A L\ tm (c) d(t) = 3 cos(8t) (d) d(t) = 6 cos(8t) A 2. An object with a mass of 25 grams is attached to the end of a coil spring and is pulled down a distance of 3 cm. from its rest position and then released. There is a damping factor of.65 grams/second. Assume that the positive direction of motion is up and the object takes 4 seconds to go one full oscillation (under simple harmonic motion). Write an equation that relates the displacement d (in centimeters) of the object from its rest position after t seconds. (a) d(t) = 3e Q.R6`/6K cos bc \d A (.R6)d 86KK te (b) d(t) = 3eQ.R6`/86 cos bc4 (.R6)d 86KK te (c) d(t) = 3e Q.R6`/6K cos bc \d A (.R6)d 86KK te (d) d(t) = 3eQ86`/(J.5) cos bc \d A 86d A(.R6) d te

$3. Sketch the graph of the following damped vibration curve: d(t) = e Q`/A\ cos (2t) (a) (b) (, e QJ/8 ) (4, e QJ ) 4 (c) (d) (, e QJ/A ) (, e QJ/8 ) 4.$

3 3. Sketch the graph of the following damped vibration curve: d(t) = e Q`/A\ cos (2t) (a) (b) (, e QJ/8 ) (4, e QJ ) 4 (c) (d) (, e QJ/A ) (, e QJ/8 ) 4. Which of the following polar coordinates describe the same point as the point L5, \ R M given in polar coordinates. (a) L5, 6\ M (b) L 5, 6\ M (c) L5, \ M (d) L 5, \ M (e) none of the above R R R R 5. Which of the following are polar coordinates of the point whose rectangular coordinates are h 2, 2 3i. (a) L2 3, A\ M (b) M (c) L4, \ M (d) L4, A\ M (e) none of the above 5 R Convert the following equation from an equation using polar coordinates to an equation using rectangular coordinates: 6 sin θ = r 5 cos θ (a) x 8 6x + y 8 5y = 0 (c) x 8 5x + y 8 6y = 0 (b) 6y = 5x (d) none of the above

4 7. Identify the graph of the following polar equation: r = 2 3 cos θ (a) (b) (c) (d) (e) (f) 8. Identify the graph of the following polar equation: r = 4sin(3θ) (a) (b) (c) (d) (e) (f) 9. If z = 4(cos 45 + i sin 45 ) and w = 2(cos 35 + i sin 35 ) are complex numbers, find the product z w (leave your answer in polar form). (a) 2(cos 80 + i sin 80 ) (b) 8(cos 80 + i sin 80 ) (c) 2(cos 0 + i sin 0 ) (d) 8(cos 45 + i sin 35 ) 20. Using De Moivre s Theorem, identify the complex number [2(cos 0 + i sin 0 )] R written in the standard rectangular form a + bi. (a) i (b) i (c) i (d) i (e) i

5 Answers:. E 2. A 3. B 4. C 5. A 6. F 7. D 8. C 9. D 0. B. A 2. C 3. A 4. B 5. D 6. C 7. C 8. F 9. B 20. A

Math 107 Study Guide for Chapters 5 and Sections 6.1, 6.2 & 6.5

Math 107 Study Guide for Chapters 5 and Sections 6.1, 6.2 & 6.5 Math 07 Study Guide for Chapters 5 and Sections.,. &.5 PRACTICE EXERCISES. Answer the following. 5 Sketch and label the angle θ = in the coordinate plane. Determine the quadrant and reference angle for