WI2012 – MATH 1430 Dept Final Review
Macomb Community College Department of Mathematics Review for the MATH 1430 Final Exam WINTER 2012
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WI2012 – MATH 1430 Dept Final Review
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WI2012 – MATH 1430 Dept Final Review
MATH 1430 DEPARTMENT REVIEW FOR THE DEPARTMENT FINAL EXAM
The Department Final Exam will have 30 multiple-choice questions. There is only ONE correct answer. The time limit for Department Final Exam is 75 minutes. All students must take the Department Final Exam on the last day of class. The student needs to purchase a scantron (Form No. X-101864-PAR-L) from MCC’s bookstore for Department Final Exam. The student will need a #2 pencil for Department Final Exam. No scratch paper may be used during the Department Final Exam. You may write in the exam packet. A scientific calculator or a TI-83 or a TI-84 graphing calculator may be used on the Department Final Exam. No qwerty keyboards, such as the TI-89. Check with your instructor to make sure you have a permissible calculator. All cell phones and electronic devices must be POWERED OFF.
THE FOLLOWING FORMULAS NEED TO BE MEMORIZED
sin
1 csc
cos
sin 2 cos 2 1
1 sec
1 tan 2 sec 2
sin( 2 ) 2 sin cos sin
2
1 cos 2
tan
tan
sin cos
cot 2 1 csc 2 cos( 2 ) cos 2 sin 2 cos
sin( ) sin cos cos sin sin( ) sin cos cos sin
1 cot
2
1 cos 2
cos( ) cos cos sin sin cos( ) cos cos sin sin
sin A sin B sin C a b c 2 2 2 b a c 2ac cos B
c 2 a 2 b 2 2ab cos C
Arc length s r
Area of a sector A
a 2 b 2 c 2 2bc cos A
1 2 r 2
ANSWERS TO REVIEW QUESTIONS ARE ON THE LAST PAGE 3
WI2012 – MATH 1430 Dept Final Review
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WI2012 – MATH 1430 Dept Final Review
1) Convert the angle 1702433 to a decimal in degrees. Round the answer to two decimal places. a) b) c) d) e)
170.24 170.41 170.42 2.98 2.97
2) If s denotes the length of the arc of a circle of radius r = 10.6 inches subtended by a central angle 60 , find the length of the arc. a) b) c) d) e)
636 inches 63.6 inches 6.36 inches 5.5 inches 11.1 inches
5 each second. If the pendulum is 60 inches long, 18 how far does its tip move each second? If necessary, round the answer to two decimal places.
3) A pendulum swings through an angle of
a) b) c) d) e)
50.51 inches 53.65 inches 54.79 inches 52.36 inches 49.50 inches
4) Convert the angle 75 to radians. Express answer as a multiple of . a)
4 11
b)
24 5
c)
5 12
d)
6 13
e)
5 24
5
WI2012 – MATH 1430 Dept Final Review
5) Find the reference angle for 225 . a) b) c) d) e)
15 30 45 60 90
6) Convert the angle a) b) c) d) e)
165 167 164 166 165
11 to degrees. Round the answer to two place values. 12
7) The blade of a windshield wiper sweeps out an angle of 135 in one cycle. The base of the blade is 12 inches from the pivot point and the tip is 32 inches from the pivot point. What area does the wiper cover in one cycle? Round answer to one decimal place. a) b) c) d) e)
1036.8 in2 169.6 in2 1206.4 in2 1376.0 in2 471.2 in2
21 2 , is the point on the unit circle that corresponds to 8) If t is a real number and P = (x, y) = 5 5 t, find the exact value of sec t . a)
5 21 21
b)
2 21 21
c)
5 2 21 2
d) e)
2 5
6
WI2012 – MATH 1430 Dept Final Review
9) A point (4, 1) on the terminal side of an angle is given. Find the exact value of csc . a)
3 17 17
b) 17 c)
15 4
d)
17 4
e) 15 10) Find the exact value of the expression cot 60 sin 45 . a)
2 3 2
b)
2 2 2
c)
2 2 3 3 6
d)
1 2 32
e)
2 33 2 6
11) Find the exact value of tan 150 cos 210 . a)
3 32 3 6
b)
2 33 6
c)
3 1 2
d)
5 3 6
e)
1 2 7
WI2012 – MATH 1430 Dept Final Review
12) Use a calculator to approximate cot a) b) c) d) e)
8
.
146.01 145.90 2.30 2.41 0.41
13) What is the domain of the sine function? a) All real numbers from 1 to 1, inclusive. b) All real numbers, except integral multiples of . c) All real numbers. d) All real numbers, except odd multiples of e) All real numbers, except x 0 .
2
.
14) What is the range of the tangent function? a) All real numbers from 1 to 1, inclusive. b) All real numbers, except integral multiples of . c) All real numbers. d) All real numbers, except odd multiples of e) All real numbers, except x 0 .
2
.
15) For what numbers is f ( ) sec not defined? a) All real numbers. b) Odd multiples of . c) Integral multiples of . d) Odd multiples of
2
.
e) Even multiples of . 16) What is the y-intercept of the function y sec x ? a) (0, 0) b) 0, 2
c) (0, 1) d) 1, 0) e) ( , 0) 8
WI2012 – MATH 1430 Dept Final Review
17) For what values of x, 0 x 2 , does sin x 0 ? a) 0, , 2 3 b) , 2 2 c) 0, 1 d) 0, 1, 2 e) – 1, 1 18) Name the Quadrant in which the angle lies if csc 0 and sec 0 . a) b) c) d) e)
I II III IV V
19) Find the exact value of cos if tan a)
10 91 91
b)
3 109 109
c)
91 3
d)
3 109 109
e)
3 91 91
20) Find the exact value of csc if cos a)
20 21
b)
21 29
c)
29 20
d)
29 21
10 and is in quadrant II. 3
20 3 and 2 . 29 2
e) None of the above. 9
WI2012 – MATH 1430 Dept Final Review
1 21) Find the amplitude for the function y 3 sin x . 2 a) 4
b) 3 c) 3 d) e)
3 3 2
22) Find the period for the function y 3 cos(5 x ) . a) 1 b) 2 c) 5 d) e)
2 5
5
23) Find the phase shift if y 4 sin 2 x . 2 a) 2 units down b) c)
2
units to the left
units to the right 4 d) 4 units up e)
4
units to the left
24) Write the equation of a sine function that has an amplitude of 3 and a period of 4 . a) y 3 sin( 2 x ) 1 b) y 3 sin x 2 c) y 3 sin(2 x ) d) y 3 sin(8 ) e) y 3 sin( 4 ) 10
WI2012 – MATH 1430 Dept Final Review
25) Find the equation for the following graph.
a) y 3 cos(10 x ) b) y 3 sin(10 x ) 1 c) y 3 cos x 5 d) y 3 cos 5
y
x
x
e) y 3 sin( x ) 5 26) Find the exact value of tan 1 ( 1) . a) b)
7 4
4
c)
4 3 d) 4 e) 1
27) Use a calculator to find the value of sin 1 a) b) c) d) e)
7 . 5
31.05 58.05 1.01 0.56 5.6
1 28) Find the exact value of cos 2 sin 1 . 4 3 a) 8 b)
7 8
c)
1 8
d)
5 8
e) 1 11
WI2012 – MATH 1430 Dept Final Review
29) Complete the identity.: a) b) c) d) e)
0 2 tan 2 1 cot sin tan 1
30) Complete the identity: a) b) c) d) e)
sin sin = 1 sin 1 sin
tan 2 3 sin tan sec =
sec csc sin tan 2 tan 2 1 cot 1
31) Complete the identity:
sin 2 1 = sin 1
a) sin b) sin 1 cos 2 c) cos d) sin 1 e) 1 32) Write sin tan 1 u as an algebraic expression in terms of u. a)
1 u 1
b)
1 u 1
2
2
u
c) d) e)
u2 1
1 u u u2 1
12
WI2012 – MATH 1430 Dept Final Review
33) Use the sum or difference identity to find the exact value of tan 285 . a)
2 3 4
b)
2 3 4
c) 2 3 d) 2 3 e) 1
34) Find the exact value of 3 3
a) b)
tan 65 tan 85 . 1 tan 65 tan 85
1 2
c) 2 d) 3 e) 1
35) Find the exact value of sin( ) if sin when
2
2 4 3 when and tan 5 2 21
.
a)
8 3 21 25
b)
8 3 21 25
c)
6 4 21 25
d)
6 4 21 25
e)
4 21 6 25 13
WI2012 – MATH 1430 Dept Final Review
2 1 36) Find the exact value of sin sin 1 cos 1 . 3 3 a)
2 3 2 10 9
b)
2 3 5
c)
2 2 10 9
d)
2 6 5
e)
2 130 9
37) Write cossin 1 u cos 1 v as an algebraic expression in terms of u and v.
1 v uv u 1 v 1 v 1 u u 1 v v u 1 u v 1
a) uv 1 u 2 b) c) d)
2
2
2
2
2
2
2
e) 1
38) Find the exact value of cos(2 ) if csc a)
4 21 25
b) c)
17 25
17 25
d) e)
5 and tan 0 . 2
4 21 25
25 17 14
WI2012 – MATH 1430 Dept Final Review
39) Find the exact value of sin( 2 ) if cos a) b)
119 169
120 169
c) d)
120 169
119 169
e)
5 13
40) Find the exact value of sin
2
if sec
17 and . 15 2
17 17
a)
b) c) d)
12 3 and 2 . 13 2
17 17 4 17
4 17 17
e) 1
41) Solve the equation sin( 4 ) a) 0, b) c) d)
4
3 on the interval 0 2 . 2
,
2 7 8 13 14 19 20 3
,
3
,
3
,
3
,
3
,
3
,
3
,
3
7 2 13 7 19 5
, , , , , , , 12 6 12 3 12 6 12 3
2 7 8 13 14 19 20 6
,
6
,
6
,
6
,
6
,
6
,
6
,
6
e) No Solution
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WI2012 – MATH 1430 Dept Final Review
42) Solve the equation csc(3 ) 0 on the interval 0 2 . 9 a) , 8 8 b)
3 5 7 ,
4
c) 0,
4
,
4
,
4
2 4 ,, 3 3
d) 0, , 2 , 3 , 4 , 6 e) No Solution 43) Solve the equation tan 1.5 on the interval 0 2 . a) b) c) d) e)
0.98 2.55 0.98, 0.98 0.98. 4.12 No Solution
44) Solve the equation cos 2 2 cos 1 0 on the interval 0 2 . a)
7 4
,
4 3 , b) 2 2 c) 2 d) e) No Solution 45) Two sides a = 7 and b = 8 of a right triangle ABC (C is the right angle) are given. Find csc B . a)
7 113 113
b)
113 8
c)
8 113 113
d)
113 7
e) 1 16
WI2012 – MATH 1430 Dept Final Review
sec 50 46) Find the exact value of . csc 40 a) 0 b) 1 c) 1 d) 90 e) 1
47) John (whose line of sight is 6 ft above horizontal) is trying to estimate the height of a tall oak tree. He first measures the angle of elevation from where he is standing as 35 . He walks 30 feet closer to the tree and finds that the angle of elevation has increased by 12 . Estimate the height of the tree rounded to the nearest whole number. a) b) c) d) e)
56 ft 61 ft 67 ft 86 ft 90 ft
48) A twenty-five foot ladder just reaches the top of a house and forms an angle of 41.5 with the wall of the house. How tall is the house? Round answer to one place value after decimal. a) b) c) d) e)
18.6 ft 18.7 ft 19.0 ft 18.8 ft 16.6 ft
49) Two hikers on opposite sides of a canyon each stand precisely 525 meters above the canyon floor. They each sight a landmark on the canyon floor on a line directly between them. The angles of depression from each hiker to the land mark are 37 and 21 . How far apart are the hikers? Round your answer to the nearest whole number. a) b) c) d) e)
2065 meters 671 meters 696 meters 1064 meters 1368 meters
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WI2012 – MATH 1430 Dept Final Review
50) Two sides and an angle of a non-right triangle are given as a = 7, b = 9, and B 49 . Solve the triangle. a) A 76.01, C 54.99, c 7.60 b) A 35.94, C 95.06, c 11.88 c) A1 1 76.01, C1 1 54.99, c1 7.60 or A2 2 103.99, C2 2 27.02, c2 12.14 d) 1 e) No Triangle 51) Two sides and an angle of a non-right triangle are given as b = 5, c = 7, and B 65 . Solve the triangle. a) C 32, A 47, a 14 b) B 33, A 82, a 12 c) C 34, A 81, a 16 d) C 65 , A 50 , a = 7 e) No Triangle 52) Three sides of a non-right triangle are given as a = 4, b = 5, and c = 6. Solve the triangle. a) A 55.8, B 41.4, C 82.8 b) A 55.8, B 82.8, C 41.4 c) A 41.4, B 55.8, C 82.8 d) A 41.4, B 34.2, C 104.4 e) No Triangle 53) Island A is 150 miles from island B. A ship captain travels 250 miles from island A and then realizes that he is off course and 160 miles from island B. What angle, in degrees, must he turn through to head straight for island B. a) b) c) d) e)
34.92 55.08 145.08 110.17 180.00
18
WI2012 – MATH 1430 Dept Final Review
54) Find the area of the triangle where A 20 , b = 11, and c = 9. a) b) c) d) e)
14.93 units2 46.51 units2 48.51 units2 16.93 units2 33.86 units2
55) Find the area of the triangle where a = 15, b = 14, and c = 17. a) b) c) d) e)
108.68 units2 102.68 units2 105.68 units2 99.68 units2 95.68 units2
3 56) The polar coordinates of a point are 3, . Find the rectangular coordinates for the point. 4
3 2 3 2 , a) 2 2 3 2 3 2 , b) 2 2 3 2 3 2 , c) 2 2 3 2 3 2 , d) 2 2 2 2 e) , 2 2
19
WI2012 – MATH 1430 Dept Final Review
57) The rectangular coordinates of a point are ( 3, 1) . Find the polar coordinates for the point.
5 a) 2, 6 b) 2, 6
c) 2, 6 5 d) 2, 6
e) 2, 6
58) The letters r and represent polar coordinates. Write the equation r cos using rectangular coordinates. a) b) c) d) e)
x2 y2 y ( x y )2 y x2 y2 x ( x y )2 x ( x y ) xy
59) The letters x and y represent rectangular coordinates. Write the equation x 2 y 2 4 x 0 using polar coordinates. a) b) c) d) e)
r cos r 2 cos 2 4 sin
r cos 2 r sin 2 r cos 0 r sin r 4 cos
20
WI2012 – MATH 1430 Dept Final Review
60) Find the equation that represents the following graph.
y
a) b) c) d) e)
r 2 3 cos r 2 3 cos r 4 5 sin r 4 5 sin None of the above.
x
61) Find the equation that represents the following graph. a) b)
3
4
c) r 30 d) r 45 e) 90
y
x
62) Write the complex number 3 4i in polar form. a) b) c) d) e)
7(cos 233.1 i sin 233.1) 7(cos 53.1 i sin 53.1) 5(cos 126.9 i sin 126.9) 5(cos 306.9 i sin 306.9) 5(cos 53.1 i sin 306.9)
63) Write the complex number 6(cos 330 i sin 330) in rectangular form. a) 3 3 3i b) 3 3 3i c) 3 3 3i d) 3 3 3i e)
3 1 i 2 2 21
WI2012 – MATH 1430 Dept Final Review
64) Given the complex numbers z 10(cos 30 i sin 30) and w 5(cos 10 i sin 10) , find zw. a) b) c) d) e)
15(cos 300 i sin 300) 50(cos 40 i sin 40) 50(cos 300 i sin 300) 15(cos 40 i sin 40) 15
65) Given the complex numbers z 10(cos 30 i sin 30) and w 5(cos 10 i sin 10) , z find . w a) b) c) d) e)
5(cos 3 i sin 3) 2(cos 3 i sin 3) 2(cos 20 i sin 20) 5(cos 20 i sin 20) 15
66) The vector v has initial position P = (6, 4) and terminal point Q = (3, 2). Find the position vector. a) v = 7i – 4j b) v = 6i – 3j c) v = 4i + 7j d) v = 3i + 6j e) v = 3i + 6j
67) If v = 3i 5j and w = 7i + 4j, find 3v – 4w. a) 19i + j b) 37i – 31j c) 17i – 10j d) 4i – j e) – 19i – 31j
22
WI2012 – MATH 1430 Dept Final Review
68) If v = 3i 5j, find v . a) b) c) d) e)
2 34 2 2 8 4
69) If v = 5i 7j and w = 3i j, find v w . a) 89 b) 39 c) 2 6 13 d) 3 e) 3 19 70) Find the unit vector having the same direction as v = 4i – 3j. a) u =
4 3 i+ j 5 5
b) u =
5 5 i+ j 4 3
4 3 c) u = i j 5 5 d) u =
5 5 i j 4 3
e) u = 1
23
WI2012 – MATH 1430 Dept Final Review
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WI2012 – MATH 1430 Dept Final Review
ANSWERS: 1) B 2) E 3) D
36) C 37) C 38) C
4) C 5) C 6) A
39) C 40) D 41) C
7) A 8) A 9) B
42) E 43) D 44) D
10) E 11) E 12) D
45) B 46) C 47) C
13) C 14) C 15) D
48) B 49) A 50) B
16) C 17) A 18) A
51) E 52) C 53) C
19) D 20) E 21) B
54) D 55) D 56) B
22) D 23) C 24) B
57) C 58) C 59) E
25) D 26) C 27) D
60) B 61) A 62) D
28) B 29) B 30) C
63) B 64) B 65) C
31) D 32) E 33) D
66) D 67) B 68) B
34) A 35) E
69) A 70) C 25
