Unformatted text preview: Math 111 Final Exam 1. In the given right triangle ඃ = 7 and ඃ = 40°. Find the hypotenuse c.
(a) 7 sec 50° (b) 7 cos 40° (d) 7 sin 50° c (c) 7 csc 40° (e) 7 sec 40° b 40° ඃ = 7 (Drawing not to scale) (f) 7 sin 40° 2. Use fundamental identities and/or the complementary angle theorem to find the value of the
234 56°
following expression: tan 55° − 234 66°
(a) 0 (b) √8
8 (c)√2 (d) 1 (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.
(d) 4>?(56°)
@AA (b) 744 tan(35°) ft.
(e) ft. BC?(56°)
@AA (c)744 cos(35°) ft.
(f) ft. 234(56°)
@AA ft. 4. Mike is trying to make a triangle with sides a, b, and c so that ඃ = 9, ඃ = 10, and ඃ = 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 )
(b) One triangle
(c) Two triangles
(e) Not enough information to tell (a) No triangles
(d) None of the above 5. Given the following triangle, find the measure of the missing side length x.
Simplify where possible.
(a) 2√3 sin 10° (b) 2√3 sin 110° (d) 2√3 cos 10° (e) 8
√ ∙ sin 10°
5 (c) 6 sin 10°
(f) 8
√ ∙ cos 110°
5 x 110° 3 #5 60° (Drawing not to scale) 6. Given the following triangle, find the measure of the missing side length x.
Simplify where possible.
(a) A 4>? 86°
4>? JKK°
JJ° (d) 4 sin L8K°M (b)
(e) A 4>? 86°
4>? 66°
A 4>? JKK°
4>? 66° (c)
(f) A 4>? 66°
4>? 86°
A 4>? 66°
4>? JKK° #6 55° 4 x 25° (Drawing not to scale) 7. Given the following triangle, use the law of cosines to find the measure
of the missing side length x. Simplify where possible.
(a) N29 − 5√2 (b) N21 + 10√2 (d) N29 − 10√2 (e) N29 + 10√2 (c) √19 2 #7 45° 5 x (Drawing not to scale) (f) 7 − 10√2 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 6 ft between the base of the ladder and the base of the slide section is 5 feet, 4 ft what is the measure of the angle that the slide makes with the ground
(the measure of ඃ in the picture to the right). Keep in mind that the ladder ඃ does n ot necessarily make a 90° angle with the ground and that the drawing is not to scale.
(a) 5
cosQJ 8 (b) 6
cos QJ R (c) 5
cosQJ A (d) S
cosQJ JR 5 ft e) none of the above 9. Find the area of the given triangle.
(a) 5 cos 20° in8
(d) 5 sin 20° in8 #9 (b) 10 cos 20° in8
(e) 10 sin 20° in8 2 in.
20° (c) 5 in8 5 in. (f) none of the above (Drawing not to scale) 10. 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) √15 yd8 (b) 3√15 yd8 (c) 9 yd8 (d) 12 yd8 (e) 12√210 yd8 (f) none of the above 11. 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) ඃ(ඃ) = 3 cos L\ ඃM
A (b) ඃ(ඃ) = 6 cos L\ ඃM
A (c) ඃ(ඃ) = 3 cos(8ඃ) (d) ඃ(ඃ) = 6 cos(8ඃ) 12. An object with a mass of 25 grams is attached to the end of a coil spring and is pulled down a
distance of 13 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.
d d .R6)
(a) ඃ(ඃ) = 13ඃ Q.R6`/6K cos bc\A − (86KK ඃe
d d .R6)
(c) ඃ(ඃ) = −13ඃ Q.R6`/6K cos bc\A − (86KK ඃe d .R6)
(b) ඃ(ඃ) = −13ඃ Q.R6`/86 cos bc4 − (86KK ඃe
d d (d) ඃ(ඃ) = −13ඃ Q86`/(J.5) cos bc\A − A(86 )d ඃe
.R6 13. Sketch the graph of the following damped vibration curve: ඃ(ඃ) = ඃ Q`/A\ cos (2ඃ)
(a) (b) 1 (2ඃ, ඃ ඃ QJ/8 1 ) 2ඃ (4ඃ, ඃ QJ ) 2ඃ ඃ 4ඃ ‐1 ‐1 (c) 1 ඃ 2ඃ (2ඃ , −ඃ QJ/8 ) ‐1 (d) 1 (2ඃ, ඃ QJ/A ) 2ඃ ‐1 \ 14. Which of the following polar coordinates describe the same point as the point L5, − R M given
in polar coordinates.
(a) L5, 6\
R M (b) L−5, 6\
R M \ (c) L5, R M \ (d) L−5, − R M (e) none of the above 15. Which of the following are polar coordinates of the point whose rectangular coordinates are
h−2, −2√3i.
(a) L2√3, A\
5 M (b) L4, @\
R M \ (c) L4, 5 M (d) L4, A\
5 M (e) none of the above 16. Convert the following equation from an equation using polar coordinates to an equation
using rectangular coordinates: 6 sin ඃ = ඃ − 5 cos ඃ
(a) ඃ 8 − 6ඃ + ඃ 8 − 5ඃ = 0
(c) ඃ 8 − 5ඃ + ඃ 8 − 6ඃ = 0 (b) 6ඃ = −5ඃ
(d) none of the above 17. Identify the graph of the following polar equation: ඃ = 2 − 3 cos ඃ
(a) (b) (d) (c) (e) (f) 18. Identify the graph of the following polar equation: ඃ = 4sin(3ඃ)
(a) (b) (c) (d) (e) (f) 19. If ඃ = 4(cos 45° + ඃ sin 45°) and ඃ = 2(cos 35° + ඃ sin 35°) are complex numbers, find the
product ඃ ∙ ඃ (leave your answer in polar form).
(a) 2(cos 80° + ඃ sin 80°)
(c) 2(cos 10° + ඃ sin 10°) (b) 8(cos 80° + ඃ sin 80°)
(d) 8(cos 45° + ඃ sin 35°) 20. Using De Moivre’s Theorem, identify the complex number [2(cos 10° + ඃ sin 10°)]R written
in the standard rectangular form ඃ + ඃඃ .
(a) 32 + 32√3ඃ (b) 6 + 6√3ඃ (c) 32√3 + 32ඃ (d) 6√3 + 6ඃ (e) 64 + 64√3ඃ A nsw e r s :
1. E
2. A
3. B
4. C
5. A
6. F
7. D
8. C
9. D
10. B
11. A
12. C
13. A
14. B
15. D
16. C
17. C
18. F
19. B
20. A ...
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 Fall '11
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 Math, Pythagorean Theorem, Cartesian Coordinate System, Cos, triangle, Polar coordinate system, Billy

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