Download Section 4.1

Jim Lambers
Math 1B
Fall Quarter 2004-05
Final Exam Practice Problems
The following problems are indicative of the types of problems that will appear on the Final
Exam, which will be given on Monday, December 6. The exercises are from the textbook, Precalculus, 5th Edition, by Barnett, Ziegler, and Byleen. As they are odd-numbered exercises, you can
find the answers at the end of this document.
In doing these problems, you should try to simulate exam conditions as closely as possible. It
is recommended that you observe the following guidelines:
• Select a quiet working environment.
• Try to complete the problems as rapidly as possible. For a useful practice exam, pre-select
problems that you have not previously attempted and see how many of them you can complete
accurately in 45 minutes.
• Scientific calculators will be allowed, so be sure to have one while attempting these problems.
• Do not use your books or notes while attempting these problems.
Section 4.1
61. Population Growth. Because of its short life span and frequent breeding, the fruit fly
Drosophilia is used in some genetic studies. Raymond Pearl of Johns Hopkins University,
for example, studied 300 successive generations of descendants of a single pair of Drosophilia
flies. In a laboratory situation with ample food supply and space, the doubling time for a
particular population is 2.4 days. If we start with 5 male and 5 female flies, how many flies
should we expect to have in:
(A) 1 week?
(B) 2 weeks?
65. Finance. Suppose $4,000 is invested at 11% compounded weekly. How much money will be
in the account in:
(A)
1
2
year?
(B) 10 years?
Compute answers to the nearest cent.
1
67. A couple just had a new child. How much should they invest now at 8.25% compounded daily
in order to have $40,000 for the child’s education 17 years from now? Compute the answer
to the nearest dollar.
Section 4.4
53. Sound. What is the decibel level of:
(A) The threshold of hearing, 1.0 × 10−12 watt per square meter?
(B) The threshold of pain, 1.0 watt per square meter?
Compute answers to 2 significant digits.
55. Sound. If the intensity of a sound from one source is 1,000 times that of another, how much
more is the decibel level of the louder sound than the quiet one?
57. Earthquakes. The largest recorded earthquake to date was in Colombia in 1906, with an
energy release of 1.99 × 1017 joules. What was its magnitude on the Richter scale? Compute
the answer to one decimal place.
Section 4.5
1-11. Solve to 3 significant digits.
1. 10x = 27.5
3. e−x = 0.0028
5. 102x+5 = 43.7
7. e4−2x = 45
9. 3x = 35
11. 5−x = 250
13-17. Solve exactly.
13. log x + log 4 = 1
15. ln 8 − ln x = 2
17. log(x + 10) + log(x − 5) = 2
19-25. Solve to 3 significant digits.
19. 2 = 1.0024x
2
21. e−0.005x = 100
23. 1, 000 = 75e0.5x
2
25. e−0.1x = 0.2
27-37. Solve exactly.
27. log x − log 5 = log 2 − log(x − 3)
29. ln x = ln(2x − 1) − ln(x − 2)
31. log(2x + 1) = 1 − log(x − 1)
33. (ln x)3 = ln x4
35. ln(ln x) = 1
37. xlog x = 100x
77. How many years, to the nearest year, will it take a sum of money to double if it is invested
at 15% compounded annually? Use the formula A = P [1 + (r/n)]nt .
Section 5.2
1. Write the value of each circular function in terms of the coordinates (a, b) of the circular point
W (x).
(A) cos x
(B) csc x
(C) cot x
(D) sec x
(E) tan x
(F) sin x
3-19 Find the exact value of each expression (if it exists) without the use of a calculator.
3. cos 0
5. sin(π/6)
7. sin(π/2)
9. tan(π/3)
11. tan(π/2)
13. sec 0
15. sec(π/4)
3
17. tan(π/4)
19. csc 0
21-25 In which quadrants must W (x) lie so that:
21. cos x < 0
23. sin x > 0
25. cot x < 0
75-79 Use the basic identities to find the values of the other five circular functions given the indicated
information.
75. cos x =
77. sin x =
79. tan x =
1
2
√
and tan x < 0
3
2
√
and cot x < 0
3 and sin x < 0
Section 5.3
1-3 Find the degree measure of each angle, keeping in mind that an angle of one complete rotation
corresponds to 360◦ .
1.
3.
1
9
3
4
rotation
rotation
9-11 Find the radian measure of each angle, keeping in mind that an angle of one complete rotation
corresponds to 2π radians.
9.
11.
1
8
3
4
rotation
rotation
13-15 Find the exact radian measure, in terms of π, of each angle.
13. 30◦ , 60◦ , 90◦ , 120◦ , 150◦ , 180◦
15. −45◦ , −90◦ , −135◦ , −180◦
17-19 Find the exact degree measure of each angle.
17. π/3, 2π/3, π, 4π/3, 5π/3, 2π
19. −π/2, −π, −3π/2, −2π
4
Section 5.4
27-31 Find the reference angle α for each angle θ.
27. θ = 300◦
29. θ = 7π/6
31. θ = −5π/3
33-47 Evaluate exactly, using reference angles where appropriate, without using a calculator.
33. tan(3π/4)
35. sin(−30◦ )
37. sec(5π/6)
39. cot 315◦
41. csc(−150◦ )
43. cos(13π/6)
45. tan(−7π/3)
47. sec(23π/4)
Section 5.5
1-5 Suppose that one of the acute angles of a right triangle is θ. Let a be the length of the
side adjacent to θ, let b be the length of the side opposite θ, and let c be the length of the
hypotenuse. Write the ratios of sides corresponding to each trigonometric function given. Do
not look back at the definitions.
1. sin θ
3. csc θ
5. tan θ
7-11 Consider the right triangle in Problems 1-5. Each ratio defines a trigonometric function of
the complement of θ (that is, the other acute angle in the triangle). Indicate which function
without looking back at the definitions.
7. b/a
9. a/c
11. b/c
5
19-25 Suppose that one of the acute angles of a right triangle is α, and the other acute angle is
β. Let a be the length of the side adjacent to β, let b be the length of the side opposite β,
and let c be the length of the hypotenuse. For the given angle and given side length, find the
lengths of the other two sides.
19. β = 17.8◦ , c = 3.45
23. α = 23◦ , a = 54
25. α = 53.21◦ , b = 23.82
Section 5.9
1-27 Find exact values without using a calculator.
1. cos−1 0
√
3. sin−1 ( 3/2)
√
7. sin−1 ( 2/2)
9. cos−1 1
11. sin−1 (1/2)
√
19. cos−1 (− 3/2)
23. sin−1 (−1/2)
Section 6.1
1-63 Verify the following identities.
1. sin θ sec θ = tan θ
3. cot u sec u sin u = 1
5.
sin(−x)
cos(−x)
7. sin α =
= − tan x
tan α cot α
csc α
9. cot u + 1 = (csc u)(cos u + sin u)
11.
13.
15.
cos x−sin x
sin x cos x = csc x −
sin2 t
cos t + cos t = sec t
cos x
= sec x
1−sin2 x
sec x
17. (1 − cos u)(1 + cos u) = sin2 u
19. cos2 x − sin2 x = 1 − 2 sin2 x
6
21. (sec t + 1)(sec t − 1) = tan2 t
23. csc2 x − cot2 x = 1
37.
1−(sin x−cos x)2
sin x
= 2 cos x
cot θ+1
csc θ
sin2 y
(1−cos y)2
39. cos θ + sin θ =
41.
1+cos y
1−cos y
=
43. tan2 x − sin2 x = tan2 x sin2 x
45.
51.
53.
csc θ
cot θ+tan θ = cos θ
1−cos A
sec A−1
1+cos A = sec A+1
sin4 w − cos4 w =
59. (sec x − tan x)2 =
63.
1+sin v
cos v
=
1 − 2 cos2 w
1−sin x
1+sin x
cos v
1−sin v
Section 6.2
11-43 Verify each identity.
11. cot π2 − x = tan x
13. csc π2 − x = sec x
33. cos 2x = cos2 x − sin2 x
cot x cot y−1
cot x+cot y
2 tan x
1−tan2 x
35. cot(x + y) =
37. tan 2x =
39.
sin(v+u)
sin(v−u)
=
cot u+cot v
cot u−cot v
cos(x+y)
sin x cos y
cot y−cot x
cot x cot y+1
41. cot x − tan y =
43. tan(x − y) =
Section 6.3
15-27 Verify each identity.
15. (sin x + cos x)2 = 1 + sin 2x
17. sin2 x = 21 (1 − cos 2x)
19. 1 − cos 2x = tan x sin 2x
7
x
1−cos x
2 =
2
θ
sin θ
cot 2 = 1−cos θ
1−tan2 u
cos 2u = 1+tan
2u
1+tan2 x
2 csc 2x = tan x
21. sin2
23.
25.
27.
Section 6.5
1-29 Find exact solutions over the indicated intervals, x a real number, θ in degrees.
1.
3.
5.
7.
9.
21.
23.
25.
27.
29.
2 sin x + 1 = 0, 0 ≤ x < 2π
2 sin x + 1 = 0, all real x
√
tan x + 3 = 0, 0 ≤ x < π
√
tan x + 3 = 0, all real x
√
2 cos θ − 3 = 0, 0◦ ≤ θ < 360◦
2 sin2 θ + sin 2θ = 0, all θ
tan x = −2 sin x, 0 ≤ x < 2π
2 cos2 θ + 3 sin θ = 0, 0◦ ≤ θ < 360◦
cos 2θ + cos θ = 0, 0◦ ≤ θ < 360◦
2 sin2 (x/2) − 3 sin(x/2) + 1 = 0, 0 ≤ x ≤ 2π
Section 7.1
1-7 Solve each triangle.
1.
3.
5.
7.
α = 73◦ , β = 28◦ , c = 42 feet
α = 122◦ , γ = 18◦ , b = 12 kilometers
β = 112◦ , γ = 19◦ , c = 23 yards
α = 52◦ , γ = 47◦ , a = 13 centimeters
Section 7.2
3-11 Solve each triangle.
3.
5.
9.
11.
α = 71.2◦ , b = 5.32 yards, c = 5.03 yards
◦
γ = 120 31 , a = 5.73 millimeters, b = 10.2 millimeters
a = 4 meters, b = 10.2 meters, c = 9.05 meters
a = 6 kilometers, b = 5.3 kilometers, c = 5.52 kilometers
8
Section 7.5
17-21 Convert the polar coordinates to rectangular coordinates to three decimal places.
17. (6, π/6)
19. (−2, 7π/8)
21. (−4.233, −2.084)
23-27 Convert the rectangular coordinates to polar coordinates with θ in degree measure, −180◦ <
θ ≤ 180◦ , and r ≥ 0.
23. (−8, 0)
25. (−5, −5)
27. (9.79, 5.13)
Answers
Section 4.1
61. (A) 76 flies
(B) 570 flies
65. (A) $4,225.92
(B) $12,002.71
67. $9,841
Section 4.4
53. (A) 0 decibel
(B) 120 decibels
55. 30 decibels more
57. 8.6
9
Section 4.5
1. 1.44
3. 5.88
5. −1.68
7. 0.0967
9. 3.24
11. −3.43
13. 5/2
15. 8/e2
17. 10
19. 86.7
21. −921
23. 5.18
25. ±4.01
27. 5
29. 2 +
31.
1
4 (1
√
+
3
√
89)
33. 1, e2 , e−2
35. ee
37. 0.1, 100
77. Approximately 5 years
10
Section 5.2
1. (A) a
(B) 1/b
(C) a/b
(D) 1/a
(E) b/a
(F) b
3. 1
5. 1/2
7. 1
√
9. 3
11. Not defined
13. 1
√
15. 2
17. 1
19. Not defined
21. Quadrant II or III
23. Quadrant I or II
25. Quadrant II or IV
√
√
√
√
75. sin x = − 3/2, tan x = − 3, cot x = −1/ 3, csc x = −2/ 3, sec x = 2
√
√
√
77. cos x = −1/ 2, tan x = 1, cot x = 1, csc x = − 2, sec x = − 2
√
√
√
79. cot x = 1/ 3, sin x = − 3/2, cos x = −1/2, csc x = −2/ 3, sec x = −2
11
Section 5.3
1. 40◦
3. 270◦
9. π/4
11. 3π/2
13. π/6, π/3, π/2, 2π/3, 5π/6, π
15. −π/4, −π/2, −3π/4, −π
17. 60◦ , 120◦ , 180◦ , 240◦ , 300◦ , 360◦
19. −90◦ , −180◦ , −270◦ , −360◦
Section 5.4
27. 60◦
29. π/6
31. π/3
33. −1
35. −1/2
√
37. −2/ 3
39. −1
41. −2
√
43. 3/2
√
45. − 3
√
47. 2
12
Section 5.5
1. b/c
3. c/b
5. b/a
7. cot(90◦ − θ)
9. sin(90◦ − θ)
11. cos(90◦ − θ)
19. a = 3.28, b = 1.05
23. b = 127, c = 138
25. a = 31.85, c = 39.77
Section 5.9
1. π/2
3. π/3
7. π/4
9. 0
11. π/6
19. 5π/6
23. −π/6
Section 6.1
1. Use sec θ = 1/ cos θ and tan θ = sin θ/ cos θ
3. Use cot u = cos u/ sin u and sec u = 1/ cos u
5. Use sin(−x) = − sin x, cos(−x) = cos x, and tan x = sin x/ cos x
7. Use tan α = 1/ cot α and csc α = 1/ sin α
13
9. Use csc u = 1/ sin u and cot u = cos u/ sin u
11. Use csc x = 1/ sin x and sec x = 1/ cos x
13. Use sin2 t + cos2 t = 1 and sec t = 1/ cos t
15. Use sin2 x + cos2 x = 1 and sec x = 1/ cos x
17. Use sin2 u + cos2 u = 1
19. Use sin2 x + cos2 x = 1
21. Use sec t = 1/ cos t, tan t = sin t/ cos t, and sin2 t + cos2 t = 1
23. Use csc x = 1/ sin x, cot x = cos x/ sin x, and sin2 x + cos2 x = 1
37. Use sin2 x + cos2 x = 1
39. Use cot θ = cos θ/ sin θ and csc θ = 1/ sin θ
41. Use sin2 y + cos2 y = 1
43. Use sin2 x + cos2 x = 1
45. Use cot θ = cos θ/ sin θ, tan θ = sin θ/ cos θ, sin2 θ + cos2 θ = 1, and csc θ = 1/ sin θ
51. Use sec A = 1/ cos A
53. Use sin2 w + cos2 w = 1
59. Use sec x = 1/ cos x and tan x = sin x/ cos x
63. Use sin2 v + cos2 v = 1
Section 6.2
11. Use cofunction identities for sine and cosine, and tan x = sin x/ cos x
13. Use sin(π/2 − x) = cos x, and sec x = 1/ cos x
33. Use cos(x + y) = cos x cos y − sin x sin y with y = x
35. Use double-angle identity for tangent, and cot x = 1/ tan x
37. Use sum identity for tangent
39. Use sum and difference identities for sine
14
41. Use cos(x + y) = cos x cos y − sin x sin y, cot x = cos x/ sin x, and tan y = sin y/ cos y
43. Use tan x = 1/ cot x
Section 6.3
15. Use sin2 x + cos2 x = 1 and sin 2x = 2 sin x cos x
17. Use cos 2x = 1 − 2 sin2 x
19. Use sin 2x = 2 sin x cos x and cos 2x = 1 − 2 sin2 x
21. Use half-angle identity for sine
23. Use half-angle identity for tangent, and cot(θ/2) = 1/ tan(θ/2)
25. Use double-angle identity for tangent, and cot 2u = 1/ tan 2u
27. Use csc 2x = 1/ sin 2x, sin 2x = 2 sin x cos x, and tan x = sin x/ cos x
Section 6.5
1. 7π/6, 11π/6
3. 7π/6 + 2kπ, 11π/6 + 2kπ, k any integer
5. 2π/3
7. 2π/3 + kπ, k any integer
9. 30◦ , 330◦
21. k(180◦ ), 135◦ + k(180◦ ), k any integer
23. 0, 2π/3, π, 4π/3
25. 210◦ , 330◦
27. 60◦ , 180◦ , 300◦
29. π/3, π, 5π/3
15
Section 7.1
1. γ = 79◦ , a = 41 ft, b = 20 ft
3. β = 40◦ , a = 16 km, c = 5.8 km
5. α = 49◦ , a = 53 yd, b = 66 yd
7. β = 81◦ , b = 16 cm, c = 12 cm
Section 7.2
3. a = 6.03 yd, β = 56.6◦ , γ = 52.2◦
◦
5. c = 14.0 mm, α = 20 32 , β = 39◦
9. α = 23.0◦ , β = 94.9◦ , γ = 62.1◦
11. α = 67.3◦ , β = 54.6◦ , γ = 58.1◦
Section 7.5
17. (5.196, 3.000)
19. (1.848, −0.765)
21. (2.078, 3.688)
23. (8, 180◦ )
√
25. (5 2, −135◦ )
27. (11.05, 27.7◦ )
16