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math_330_final_exam_review__3_.pdf

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Final Exam Review
Math 330 Trigonometry
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use “Definition I” with (x,y) and r to find the exact value below.
1) csc 240°
A) – 2
3
B)
3
2
3
3
1)
C) -2
D) 2
Use the fundamental identities to find the value of the trigonometric function.
2) Find sin ?, given that cos ? = 2 and ? is in quadrant IV.
7
A) – 3
5
B) –
2
7
2
C) – 3
5
7
2)
D) -3
5
Use “Definition II” (SOH-CAH-TOA) to find the exact value below. Write your answer as a fraction in lowest terms.
3)
3)
20
12
16
Find sin A.
A) sin A = 5
4
B) sin A = 4
5
C) sin A = 4
3
D) sin A = 3
5
Find all values of ?, if ? is in the interval [0, 360°) and has the given function value.
4) cos ? = –
3
4)
2
A) 210° and 330°
5) sec ? = – 2
A) 45° and 225°
B) 60° and 120°
C) 60° and 300°
D) 150° and 210°
B) 45° and 315°
C) 135° and 225°
D) 225° and 315°
5)
Solve the right triangle. If two sides are given, give angles in degrees and minutes.
6)
A = 41.3°, b = 2.5 m
Round side lengths to one decimal place.
A) B = 48.7°; a = 1.1 m; c = 4.0 m
C) B = 48.7°; a = 2.2 m; c = 3.3 m
6)
B) B = 48.7°; a = 4.0 m; c = 4.7 m
D) B = 48.7°; a = 1.1 m; c = 2.7 m
Solve the problem.
7) A ship travels 82 km on a bearing of 24°, and then travels on a bearing of 114° for 162 km. Find
the distance from the starting point to the end of the trip, to the nearest kilometer.
A) 75 km
B) 244 km
C) 33 km
D) 182 km
Use “Definition III” with the unit circle to find the exact value below.
8) csc 5″
3
A) –
B) – 1
2
3
C) – 2
7)
8)
3
3
D) –
2
Find the length of an arc intercepted by a central angle ? in a circle of radius r. Round your answer to 1 decimal place.
9) r = 115.19 in.; ? = 195°
9)
A) 784.1 in.
B) 124.8 in.
C) 196.0 in.
D) 392.0 in.
Find the area of a sector of a circle having radius r and central angle ?. If necessary, express the answer to the nearest
tenth.
10) r = 19.0 m, ? = 20°
10)
2
2
2
2
A) 63.0 m
B) 126.0 m
C) 0.6 m
D) 3.3 m
Find the exact values of s in the given interval that satisfy the given condition.
11) [0, 2″); sin s =
A)

3
3
11)
2
B)
” , 2″
3 3
C)
” , 3″
4 4
D)

4
Solve the problem.
12) A wheel is rotating at 5 radians/sec, and the wheel has a 83-inch diameter. To the nearest foot,
what is the speed of a point on the rim in ft/min?
A) 1033 ft/min
B) 1038 ft/min
C) 1043 ft/min
D) 1028 ft/min
12)
13) An object is spinning around a circle with a radius of 20 centimeters. If in 17 seconds a central
angle of 1 radian has been covered, what is the linear speed of the object?
18
A)
10
cm per sec
153
B)
C)
20
cm per sec
153
D) 20 cm per sec
13)
170
cm per sec
9
For #14-16, graph one period of the function, labeling key values. (Be able to also identify the period, domain and
range)
14) y = 4 + 2 sin(x – “)
14)
y
7
6
5
4
3
2
1
p
2
p
3p
2
2p
x
A)
B)
y
y
7
7
6
6
5
5
4
4
3
3
2
2
1
1
p
2
p
3p
2
2p
x
C)
p
2
p
3p
2
2p
p
2
p
3p
2
2p
x
D)
y
y
7
7
6
6
5
5
4
4
3
3
2
2
1
1
p
2
p
3p
2
2p
x
x

15) y = 3 cos x +
4
15)
y
3x
A)
B)
y
y
3x
C)
3x
D)
y
y
3x
3x
16) y = 1 tan 2x
4
16)
y
p
x
A)
B)
3
y
y
p
x
p
x
-3
-3
C)
D)
3
y
3
p
y
x
p
-3
x
-3
Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
1 + tan 2 x
17)
17)
sec x
A) sec x
B) -sec x
C) csc x
D) 1
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Verify that each equation is an identity.
18) cot2x = (csc x – 1)(csc x + 1)
18)
19)
sec ? – 1 = tan ?
tan ?
sec ? + 1
19)
20)
1 – cot2? + 1 = 2 sin2?
1 + cot2?
20)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the exact value of the expression using the provided information.
21) Find cos(s + t) given that cos s = – 1 , with s in quadrant III, and cos t = – 3 , with t in quadrant III.
2
5
A)
3 +4 3
10
B)
-3-4
10
3
C)
3-4 5
10
D)
3-4 3
10
22) Find sin(s – t) given that cos s = 1 , with s in quadrant I, and sin t = – 1 , with t in quadrant IV.
3
2
3 +2
6
A)
2
B)
2
6 +1
6
C)
2
6-1
6
3-2
6
D)
6+
4
2
23)
B)
24
6
Use a sum or difference identity to find the exact value.
24) tan 75°
A) 3 – 2
B) – 3 + 2
C)
64
2
D)

64
2
24)
C)
3 +2
D) –
3-2
25) sin 11″
12
A)

22)
2
Use Identities to find the exact value.
23) cos (-165°)
A)
21)
25)
64
2
B)
64
2
C)
6+
4
2
D)

6+
4
2
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Verify that the equation is an identity.
26) tan ” + x = -cot x
2
26)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use identities to find the indicated value for each angle measure.
20
27) sin ? =
, cos ? > 0
Find cos(2?).
29
A) –
41
841
43
B) 841
Find the exact value by using a half-angle identity.
28) cos 75°
A) – 1
2- 3
B) – 1
2+ 3
2
2
27)
41
C) 841
D)
840
841
1
C)
2
1
D)
2
28)
2+
3
2-
3
Find the exact value of the expression using the provided information.
29) Find sin ? , given that cos ? = 1 and ? terminates in 0 < ? < 90°. 2 4 10 4 A) 8-2 4 B) 15 6 C) 4 29) 8 +2 4 D) Find the exact value of the real number y. 30) y = csc-1 (2) A) - " 6 15 30) B) - " 3 C) " 3 D) " 6 Give the degree measure of ?. 3 31) ? = arctan 31) 3 A) 150° B) -30° C) 330° D) 30° Give the exact value of the expression. 1 32) cos arcsin 4 A) 4 15 15 33) cos arcsin A) 56 65 B) 32) 15 2 C) 2 15 15 D) 15 4 5 3 + arccos 13 5 33) B) 48 65 C) 72 65 D) 16 65 Solve the equation for exact solutions over the interval [0, 2"). 34) sin2x + sin x = 0 34) " 2" A) 0, ", , 3 3 " 5" B) 0, ", , 3 3 3" C) 0, ", 2 4" , 5" D) 0, ", 3 3 Solve the equation (x in radians and ? in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. 35) 4 sin2 x - 1 = 0 35) A) " + n", 5" + n" 6 6 B) " + n", 5" + n" 3 6 C) " + n" 3 D) " + n", " + 2n" 6 2 Solve the equation in the interval [0°, 360°). Give solutions to the nearest tenth, if necessary. 36) 3 sin 2? - sin ? - 4 = 0 A) {270°} B) {90°} C) {180°} Solve the equation for solutions in the interval [0, 2"). 37) sin x cos x = 1 2 37) A) {0} B) " , 5" 4 4 " 5" C) 0, , ", 4 4 D) " , " , 2" , 7" , 7" , 13" , 5" 12 6 3 12 6 12 3 Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree. 38) 3 sec 2? = 2 A) {0°, 120°, 180°, 240°} B) {105°, 165°, 285°, 345°} C) {30°, 90°, 150°, 270°} D) {15°, 165°, 195°, 345°} Solve the equation for exact solutions. 39) -sin-1(4x) = " 4 A) 2 8 B) - 36) D) {0°} 38) 39) 2 8 C) - 2 2 D) {0} Solve the triangle. Round to the nearest tenth when necessary or to the nearest minute as appropriate. 40) 40) 87.7 ft A) B = 37.2°, b = 273 ft, c = 337.6 ft C) B = 36.8°, b = 270.5 ft, c = 337.6 ft B) B = 37.2°, b = 337.6 ft, c = 273 ft D) B = 37.2°, b = 28.2 ft, c = 22.9 ft Solve the problem. 41) Two tracking stations are on the equator 122 miles apart. A weather balloon is located on a bearing of N 32°E from the western station and a bearing of N 12°E from the eastern station. How far, to the nearest mile, is the balloon from the western station? Round to the nearest mile. A) 358 mi B) 248 mi C) 239 mi D) 349 mi 41) Find the missing parts of the triangle. Round to the nearest tenth when necessary or to the nearest minute as appropriate. 42) C = 112.9° 42) a = 8.4 km b = 11.3 km A) c = 22.3 km, A = 26°, B = 41.1° B) c = 16.5 km, A = 28°, B = 39.1° C) c = 19.4 km, A = 30°, B = 37.1° D) No triangle satisfies the given conditions. Vector v has the given magnitude and direction. Find the horizontal or vertical component of v, as indicated, if ? is the direction angle of v from the horizontal. Round to the nearest tenth when necessary. 43) a = 46.7°, |v| = 86.5; Find the horizontal component of v. 43) A) 59.3 B) 122.3 C) 63 D) 3.7 Find the component form of the indicated vector. 44) Let u = 9, -3 , v = -1, 8 . Find -6u + v. A) -55, 26 B) -53, 10 44) C) -48, 5 Find the angle between the pair of vectors to the nearest tenth of a degree. 45) 4, 5 , -4, -1 A) 152.7° B) 142.7° C) 61.4° D) -36, 7 45) D) 71.4° Write the complex number in rectangular form. " " 46) 8 cos + i sin 6 6 A) 1 + 3i 4 4 B) 3 + 1i 4 4 46) C) 4 3 + 4i D) 4 + 4 3i Write the complex number in trigonometric form r(cos ? + i sin ?), with ? in the interval [0°, 360°). 47) 4 3 - 4i A) 8(cos 30° + i sin 30°) B) 8(cos 330° + i sin 330°) C) 8(cos 60° + i sin 60°) D) 8(cos 300° + i sin 300°) 47) Answer Key Testname: MATH 330 FINAL EXAM REVIEW 1) A 2) C 3) B 4) D 5) C 6) C 7) D 8) C 9) D 10) A 11) B 12) B 13) A 14) C 15) B 16) B 17) A 18) cot2 x = csc2 x - 1 = (csc x - 1)(csc x + 1). 19) 20) sec ? - 1 = sec ? - 1 · sec ? + 1 = sec2 ? - 1 tan2 ? = = tan ? tan ? tan ? sec ? + 1 tan ?(sec ? + 1) tan ?(sec ? + 1) sec ? + 1 2 1 - cot2? + 1 = 1 - cot2? + 1 = 1 - cot ? + 1 = sin2? 1 + cot2? csc2? csc2? csc2? cos2? sin2? 1 sin2? + 1 = sin2? - cos2? + (sin2? + cos2?) = 2 sin2? 21) D 22) B 23) D 24) C 25) B 26) tan " + x = sin (("/2) + x) = sin ("/2) cos x + sin x cos ("/2) = 1 · cos x + sin x · 0 = -cot x. 2 cos (("/2) + x) cos ("/2) cos x - sin ("/2) sin x 0 · cos x - 1 · sin x 27) C 28) D 29) C 30) D 31) D 32) D 33) D 34) C 35) A 36) A 37) B 38) D 39) B 40) A 41) D 42) B Answer Key Testname: MATH 330 FINAL EXAM REVIEW 43) A 44) A 45) B 46) C 47) B ... Purchase answer to see full attachment

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