MATH1060: Final Exam Practice Problem Answers
The following are practice problems for the first exam. Any problems that do not have answers are
directly from the practice problems from one of the Midterms.
1. Convert the following angles to radians:
35π
36
2
π
(b) 10π ◦ Answer:
18
π
◦
(c) 36 Answer:
5
(a) 175◦ Answer:
2. Convert the following angles to degrees:
π
Answer: 9◦
20
540 ◦
(b) 3 Answer:
π
11π
(c)
Answer: 165◦
12
(a)
3. Find an angle coterminal with
9π
107π
such that 0 ≤ θ < 2π: Answer:
7
7
4. Find the area of a sector of a circle of radius 7 with central angle
5. Find the length of an arc cut out by a central angle of
5π
245π
. Answer:
6
12
5π
5π
in a circle of radius 3. Answer:
6
2
6. Be able to fill out a unit circle without thinking.
7. Take a right triangle with side lengths 7, 24, and 25, and let θ be the angle opposite the side
of length 24. Find:
(a) sec θ Answer:
25
7
(b) cos(90◦ − θ) Answer:
(c) sin θ Answer:
24
25
24
25
(d) cot(90◦ − θ) Answer:
24
7
8. You are skiing down a mountain with a vertical height of 1500 feet. The distance from the
top of the mountain to the base is 3000 feet. What is the angle of elevation from the base to
the top of the mountain. See Midterm solutions
p
9. If sin θ = 2/3, find cos θ using one of the Pythagorean identities.See Midterm solutions
1
10. Show that
sin θ
cos θ
+
= csc θ sec θ.
cos θ
sin θ
11. Show that sec2 θ + csc2 θ − tan2 θ − cot2 θ = 2.
√
2
and θ is in quadrant IV. Find θ, then find the value of all six standard
12. Suppose cos θ =
2
−π
trig functions at θ. Answer: θ =
or anything coterminal to it.
4
√
13. Suppose tan θ = − 3 and θ is in quadrant II. Find θ, then find the value of all six standard
2π
trig functions at θ. Answer: θ =
3
14. Graph y = 5 sin(2x).
1
15. Graph y = − cos(3x + 6) − 1.
2
16. Sketch the graph of the following functions:
(a) y = csc(x)
(b) y = arctan(x)
17. Evaluate the following expressions:
√
π
3
Answer:
(a) arcsin
2
3
(b) cos−1 (−1) Answer: π
√
3
π
−1
(c) sin
Answer:
2
3
√
−π
(d) arctan − 3 Answer:
3
√
2
π
(e) arccos
Answer:
2
4
π
−1
(f) tan (1) Answer:
4
18. A motorcycle is moving at 110 miles per hour. The radius of its wheel is 10 inches. What
is the angular velocity of the wheel in revolutions per second? Answer: 30 revolutions per
second
19. The earth makes one complete revolution around the sun every 365 days. The average distance
from the earth to the sun is 92.96 million miles. What is the linear speed of the earth hurtling
through space? Answer: 66, 676 miles per hour
20. The sun is 25◦ above the horizon. If Michelle is 5 feet tall, how long is her shadow? Answer:
10.7 feet
21. An airplane is 150 miles north and 100 miles east of the airport. The pilot wants to fly directly
to the airport. What bearing should be taken? Answer: N 33.6◦ E
2
22. Identify each of the following expressions as one of the standard trigonometric functions:
(a) sec β csc β − cot β
cos γ
(b)
1 − sin2 γ
23. Simplify the following expressions:
(a) csc4 α − cot4 α
(b) −1 + 2 sin2 δ + sin4 δ
(c) sin2 ζ + 3 cos ζ + 3
(d) sin tan + cos (e) tan ι −
sec2 ι
tan ι
24. Use the trigonometric substitution x = 5 tan λ to simplify the expression
5 sec λ
p
x2 + 25 Answer:
25. Verify the following identities:
(a) cos2 α − sin2 α = 1 − 2 sin2 α
1
1
+
= tan γ + cot γ
(b)
tan γ
cot γ
(c) cos2 θ + cos2 ( π2 − θ) = 1
x
(d) tan(sin−1 x) = √
1 − x2
26. Find all solutions to the following trigonometric equations:
(a) 4 cos2 φ − 1 = 0
(b) csc ν + cot ν = 1.
(c) tan(3η) − 1 = 0
27. Find the exact value of each of the following expressions:
(a) cos(5π/12)
(b) tan(165◦ )
(c) cos(17◦ ) cos(13◦ ) − sin(17◦ ) sin(13◦ )
π
π
π
π
(d) sin
cos + sin cos
12
4
4
12
(e) cos 75◦ + cos 15◦
28. Find the exact solutions to the following trigonometric equations in the interval [0, 2π)
(a) cos 2χ − cos χ = 0
(b) 4 − 8 sin2 µ = 0
3
(c) sin
β
+ cos β = 1
2
29. Use the power reducing formula to rewrite the expression sin2 x cos2 x in terms of the first
1 − cos(4x)
power of cosine. Answer:
8
30. Use the sum-to-product or product-to-sum formula to rewrite each expression:
(a) sin 5γ sin 3γ
(b) cos 6δ + cos 2δ
r
31. Use the half-angle formula to simplify
32. If sin u =
1 − cos 14x
.
2
7
and u lies in the second quadrant, find cos(u/2).
25
33. Use any means you like to solve the triangle with α = 24.3◦ , γ = 54.6◦ , and c = 10.3.
34. Use any means you like to solve the triangle with β = 63.2◦ , γ = 47.6◦ and b = 12.2.
35. Use any means you like to solve the triangle with β = 100◦ , b = 14, and c = 19.
36. Use Heron’s Formula to find the area of a triangle with side lengths 7, 8, and 9.
37. Use any means you like to solve the triangle with side lengths 7, 8, and 9.
38. Write the vector, ~v , with initial point (4, 5) and terminal point (5, −1) in standard form.
Answer: h1, −6i
39. Write the vector from the last question as a linear combination of the standard unit vectors
ı̂ and ̂. Answer: ı̂ − 6̂
√
40. Compute the magnitude of the vector, ~v , from the last question. Answer: k~v k = 37
41. Let ~v = h2, −7i and w
~ = h−2, 2i. Write 2~v + 3w
~ in standard form. Answer: h−2, −8i
42. Let ~u = h3, 7i and ~v = h1, 0i. Find ~u · ~v . Answer: 3
43. Find the angle between ~u and ~v from the previous problem. Answer: 66.8◦
44. Find proj~v ~u and proj~u ~v where ~u and ~v are the same vectors used in the previous problem.
21
9
Answer: proj~v ~u = h3, 0i and proj~u ~v = h √ , √ i
58
58
45. Are 3ı̂ + 17̂ and −10ı̂ + 3̂ orthogonal. Answer: No
46. Decompose 7ı̂ − 5̂into two orthogonal
vectors, one of which is a scalar multiple of 2ı̂ + 4̂.
−12
20
−12
20
Answer: 7ı̂ − 5̂ = √ ı̂ − √ ̂ + (7 − √ )ı̂ − (5 + √ )̂
18
18
18
18
√ arctan(−1/2)i
47. Convert 4 − 2i into trig form. Answer: 18e
4
48. Convert 7e
3πi/4
√
√
7 2 7 2
into standard form. Answer: −
+
i
2
2
49. Let ζ1 = 8e3πi/4 and ζ2 = 2e2πi/3 . Find ζ1 ζ2 . Answer: 16e17πi/12
50. Find
ζ1
. Answer: 4eπi/12
ζ2
51. Compute ζ23 , and write your answer in standard form. Answer: 8
52. Find all the 4-th roots of ζ1 . Write your answers in trigonometric form. Answer: 23/4 e3πi/16 , 23/4 e11πi/16 , 23/4 e19πi/16
√ 3π
53. Convert the point (−5, 5) into polar form. Answer: (5 2,
)
4
!
√
−7 3 −7
7π
into rectangular form. Answer:
,
54. Convert the point 7,
6
2
2
55. Convert the equation r = 4 sin θ into rectangular form. What shape is the graph of this
equation? Answer: x2 + y 2 = 4y or x2 + (y − 2)2 = 4. It’s a circle
56. Convert the equation y = 7x into polar form. Answer: θ = tan−1 (7)
5