Practice Questions for Midterm 2 - Math 1060Q - Fall... The following is a selection of problems to help prepare...

Practice Questions for Midterm 2 - Math 1060Q - Fall 2013
The following is a selection of problems to help prepare you for the second midterm exam. Please note the
following:
• anything from Module/Chapter 4 is fair game. That is, anything related to trigonometry could be on
there.
• there may be mistakes – email [email protected] if you find one.
• the distribution of problems on this review sheet does not reflect the distribution that will be on the exam.
• learning math is about more than just memorizing the steps to certain problems. You have to understand
the concepts behind the math. In order to test your understanding of the math, you may see questions
on the exam that are unfamiliar, but that rely on the concepts you’ve learned. Therefore, you should
concentrate on learning the underlying theory, not on memorizing steps without knowing why you’re
doing each step. (Some examples of “unfamiliar” problems can be found in the sample problems below, to
give you an idea.)
This section of the course focuses on trigonometric functions. These are certain types of
functions that have to do with angles and ratios. We have two units of measurement for angles:
degrees and radians. Although almost everything we do in this class is in terms of radians, it’s
good to be able to convert to degrees and vice versa.
1. Convert 30◦ to radians.
2. Convert
π
2
radians to degrees.
3. Convert 187◦ to radians.
4. Convert
13π
7
radians to degrees.
5. Which of the following angles correspond to the same spot on the unit circle?
π 5π 9π −7π
4, 4 , 4 , 4
6. What is an angle in the range [π, 2π] that corresponds to the angle − π6 ?
7. What is an angle in the range [−π, 0] that corresponds to the angle
5π
3 ?
The unit circle is important to know well for trigonometry, since we use it to define the
trigonometric functions. Do you understand the unit circle?
7π 7π
π
5π
8. Draw the unit circle. Label the angles 0, π6 , π3 , 3π
4 , π, 6 , 4 , 2π, − 4 , − 3 . Label the coordinates of the
points on the unit circle that correspond to those angles.
9. If a point P (t) on the unit circle has coordinates (−3/5, 4/5), then what are the coordinates of:
(a) P (t + π)
(b) P (t + π2 )
(c) P (− π2 )
(d) P (t + 100π)
(e) P (−t)
Note: you don’t need to find what t is!
10. Find an angle in [0, 2π) that is “coterminal” with the angle
65π
6 .
11. Find an angle in [0, 2π) that is “coterminal” with the angle
14π
5 .
12. Find the “reference angle” for the angle
7π
4 .
13. Find the “reference angle” for the angle
5π
6 .
14. Find the “reference angle” for the angle
17π
5 .
Do you understand what the trigonometric functions are? How to calculate them, both using
triangles or the unit circle?
15. Define sine, cosine, tangent, cosecant, secant, cotangent. What are their domain and range?
16. What is sin(30◦ )?
17. What is csc( 5π
6 )?
18. If t =
20π
3 ,
what are sin(t), csc(t), and cot(t)?
19. If cot(t) = 1 and t is in the interval [π, 2π], then what is sin(t)?
20. If cos(t) = − 21 and t is in the interval [π, 2π], then what is tan(t)?
21. Given that cos(θ) =
2
7
and θ is in quadrant IV, find sin(θ).
22. Given that tan(θ) = − 35 and θ is in quadrant II, find csc(θ).
23. Use the triangle below to answer these questions.
(a) Say α =
(b) Say β =
π
4
π
6
and BC = 8. What is AB?
and BC = 20. What is AC?
(c) Say AB = 5 an BC = 10. What is α?
What about going “backwards?” If you know the values of trig functions, can you work
backwards to find out what the angle might be?
√
24. If sin(t) =
2
2 ,
what might t be? Give all possible solutions in [−π, π].
25. If sin t = 1, what might t be? Give all possible solutions in [0, 2π].
26. If cos t = 12 , what might t be? Give all possible solutions in [0, 2π].
27. If tan t = −1, what might t be? Give all possible solutions in [0, 2π].
28. Find all values of t in the interval [0, 2π] such that sin t = 0.
29. Find all values of t in the interval [0, 2π] such that csc t = 2.
√
30. Find all values of t in the interval [0, 2π] such that cot t = 3.
Like any other function, we sometimes want to graph trigonometric functions. Can you? (We
won’t worry about really complicated graphs with lots of transformations, like −5 sin(3x − π6 ). But
simpler ones, like below, you could have to graph.
31. Graph sin(x), cos(x), and tan(x).
32. Graph f (x) = −3 sin(x).
33. Graph g(x) = cos(x − π2 ). Can you find another function that has this same graph?
34. Graph h(x) = 2 sin(4x).
35. Where does csc(x) have a vertical asymptote?
36. Write an equation for this graph.
A big thing needed in calculus is solving trigonometric equations, or in other words, solving
equations that just happen to involve trigonometric functions. Here are a bunch to work on.
37. Find all solutions to 2 sin(x) + 1 = 0 in [0, 2π].
38. Find all solutions to 3 cos x = 3 in [0, 2π].
39. Find all solutions to 3 sin x − 4 = sin x − 2 in [0, 2π].
40. Find all solutions to 4 cos2 x − 1 = 0 in [0, 2π].
41. Find all solutions to cos(2x) =
1
2
in [0, 2π].
42. Find all solutions to tan2 (x) = 3 in [0, 2π].
√
43. Find all solutions to sin x + 2 = − sin x in [−π, π].
44. Find all solutions to 2 cos(3x − 1) = 0 in [0, π].
45. Find all solutions to sec x = 2 cos x in [0, 2π].
46. Find all solutions to 2 sin2 x − sin x − 1 = 0 in [0, 2π].
47. Find all solutions to sin x cos x + cos x = 0 in [0, 2π].
48. Find all solutions to sin(2x) = − cos(2x) in [0, 2π].
Sometimes you need to use a trigonometric identity to help solve an equation. You’ll need to
use trig identities in the following problems. A hint is given for each one.
49. Find all solutions to 2 cos2 x + 3 sin x = 3 in [0, 2π).
√
50. Find all solutions to sin(2x) = 3 cos x in [0, 2π).
√
51. Find all solutions to 2 2 sin(2t) − 2 tan(2t) = 0 in [0, 2π).
52. Find all solutions to 2 cot2 x + csc2 x − 2 = 0 in [0, π].
53. Find all solutions to cos(2x) = cos x in [0, 2π].
Just like with other functions we’ve studied, trigonometric functions have inverses. Well,
partial inverses (since trig functions aren’t one-to-one). Do you know what the inverse trig
functions are? Do you know their domain and range? Can you compute them for some values?
Can you graph them?
54. What is arccos 12 ?
√
55. What is arcsin −2 2 ?
56. What is arctan 0?
57. What is sin−1 (sin( 3π
2 ))?
58. What is tan(arctan(−3))?
59. What is arccos(cos(− 4π
9 ))?
60. What is tan(sin−1 (
√
2
2 )?
5
61. What is tan(arccos( 13
))?
62. Express a solution to the equation tan(x − 3) = 5 using inverse trigonometric functions.
63. Express a solution to the equation sin(4x) + 1 =
2
3
using inverse trigonometric functions.
64. Graph the function cos−1 (2x).
65. Graph the function arctan(x) − π2 .
66. Graph the functions csc−1 (x) and
1
csc(x) .
Can you apply your knowledge to real-world applications?
67. How long a ladder do you need if you want to reach a window that’s 20 feet off the ground? You’re on
uneven terrain, so leaning a ladder any steeper than 30 degrees is unsafe.
68. An equilateral triangle is circumscribed around a circle of radius 5 (thus, the center of the triangle and
the center of the circle are at the same point). What is the area of the triangle?
69. Car A drives 40 miles north and then 28 miles east. Car B drives 20 miles north, and then 3 miles west.
They start from the same point.
(a) What is the distance between the cars?
(b) Let’s say the starting point of both cars is the origin, and east is the x-axis, north is the y-axis,
etc. What is the angle between Car A and the x-axis? (Expressed in radians or using inverse trig
functions if necessary.) What is the angle between the two cars?
70. A rope hanging straight down from a pole is 4 feet longer than the pole. When it’s stretched taut, it hits
the ground 8 feet away from the pole. How tall is the pole?
71. A plane is flying at an altitude of 36,000 feet. Looking down, the pilot sees a ship at an angle of depression
of 30 degrees. He sees a submarine at an angle of depression of 60 degrees in the same direction. How far
apart are the ship and submarine?
Here are a few additional problems that didn’t make it into any of the above sections.
72. Calculate sin( π3 − π4 ) without using a calculator.
73. Simplify the expression using identities as needed: (1 − 2 cos2 x + cos4 x).
74. Calculate cos( 7π
8 ) without using a calculator.
75. Simplify
3
tan2 x+1 .
76. Rewrite the product as a sum or difference: sin(7t) cos(3t).
77. True or false? 3 cos x + 3 sin x tan x = 3 sec x.
78. True or false?
3 cot3 x
csc x
= 3 cos x(sec2 x − 1)
79. Show work to prove that
3 sec θ−3
1−cos θ
= 3 sec θ.