Name:
Math 1060
Final Exam Review
1. Complete the following table:
θ (radians)
θ (degrees)
sin(θ)
cos(θ)
tan(θ) sec(θ)
csc(θ)
cot(θ)
2π
3
7π
3
π
6
− 5π
4
135◦
450◦
π
3π
2
√
2
2
√
−
2
2
2. Place all of the angles from problem 1 on a unit circle.
3. For each of the following angles, find the complementary angle and the supplementary angle assuming they exist.
(a) 60◦
4. Suppose sin(θ) =
(b)
5
13
2π
3
(c)
3π
8
and θ is in the third quadrant. Find cos(θ).
5. My giant novelty watch has a 2 inch long hour hand. How far does the tip of it
move between 6:00pm and 10:00pm?
6. You are standing 500 feet from Paul Bunyan. The angle of elevation to the top of
his head is 30◦ . How tall is Paul?
Final Exam Review
Math 1060
7. For each of the following functions, find the amplitude, period, vertical shift, and
phase shift, and graph at least two periods of the function.
(a) y = sin(x − π)
(c) y = 4 cos x +
(b) y = cos(π − x)
(d) y = sin
x
2
−
π
4
π
2
+4
+1
8. Graph the following functions:
π
4x
(a) y = 6 sin
+
π
2
−3
(c) y = 2 tan(πx) + 2
(b) y = − csc(4x − π)
(d) y = 2 sec(x + π)
9. You are riding a ferris wheel. It takes 150 seconds for the ferris wheel to go all the
way around once. You start on the ground; at the top of the ferris wheel you are
100 feet off the ground. Consider the function h(t), your height off the ground
after t seconds.
(a) Find the amplitude, period, phase shift, and vertical shift for the function h(t).
(b) Write down a rule for the function h(t) and graph it.
10. Wonder Woman is flying her invisible jet over you. Her altitude is 20 miles. You
are looking at her through your death ray scope. If the angle of elevation to Wonder Woman from your telescope is θ, graph the horizontal distance d(θ) from you
to Wonder Woman over the interval 0 < θ < π2 .
11. Compute:
√
(a) arcsin
2
2
(b) arctan
√
3
(c) arccos cos
3π
2
12. Wonder Woman is flying her invisible jet over you. Her altitude is 20 miles. You
are looking at her through your death ray scope. If the horizontal distance from
you to Wonder Woman is x, graph the angle of elevation a(x) from you to Wonder
Woman over the interval 0 < x < π4 .
13. Prove the following identities:
(a)
tan x+cot y
tan x cot y
= tan y + cot x
14. Suppose sin u =
5
13
and cos v = − 53 . Compute:
(b) cos(u − v)
(a) sin(u + v)
15. Compute sin
(b) (1 + sin(θ))(1 + sin(−θ)) = cos2 (θ)
π
12
sin
11π
12
.
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Final Exam Review
Math 1060
16. Solve the following equations:
(a) 4 cos2 x − 1 = 0
(c) cos(2x)(2 cos x + 1) = 0
(b) 12 sin2 x − 13 sin x + 3 = 0
17. Suppose ~u = h1, −1i and ~v = h0, 2i. Compute and draw:
(a) 2~u + ~v
(b) ~v − ~u
(c) 3~v + 6~u
18. Find a unit vector in the direction of w
~ = 3ı̂ − 4̂.
19. My friends and I are playing three-way free-for-all tug-of-war. We are pulling
on ropes with forces of 70 pounds, 50 pounds, and 100 pounds, with angles of 0◦ ,
135◦ , and 270◦ , respectively. Find the direction and magnitude of the total force.
20. I am trying to take two dogs for a walk. We are walking east. Suddenly, each dog
sees a different cat and runs in a different direction. If one bolts 60◦ north of east,
pulling on his leash with 100 pounds of force, and one bolts 60◦ south of east,
pulling on her leash with 100 pounds of force, how hard do I have to pull on the
leash to keep them in place? (Hint: draw a diagram!)
21. Suppose ~u = h1, −1i and ~v = h0, 2i. Compute:
(a) k~v k
(d) A unit vector in the direction of ~v
(b) ~u · ~v
(e) The component of ~v in the direction
of ~u
(c) A unit vector in the direction of ~u
22. The other day I parked my awesome pickup truck on a 60◦ incline. If my truck
weighs 6500 pounds, how much force did the brakes need to apply in order to prevent the truck from rolling downhill?
–3–