Precalculus - Practice Exercises Midterm Exam - Part 2

Precalculus - Practice Exercises
Midterm Exam - Part 2
Provided by LE Math Lab
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Graph the function.
1) y = cot ( x)
A)
B)
C)
D)
Answer: C
1
2) y = tan x -
4
A)
B)
C)
D)
Answer: C
Solve the problem.
3) What is the y-intercept of y = tan x?
A)
2
B) 0
C) 1
Answer: B
2
D) none
4) What is the y-intercept of y = csc x?
A) 1
B)
C) 0
2
D) none
Answer: D
5) For what numbers x, -2
A) -2 , - , 0, , 2
x
2 , does the graph of y = cot x have vertical asymptotes?
3
3
,- , ,
B) -2, -1, 0, 1, 2
C) D) none
2
2 2 2
Answer: A
Graph the function.
6) y = sec x +
2
A)
B)
3
C)
D)
Answer: C
7) y = -csc x
A)
B)
4
C)
D)
Answer: A
Solve the problem.
8) A rotating beacon is located 9 ft from a wall. If the distance from the beacon to the point on the wall where the
beacon is aimed is given by
a = 9|sec 2 t|,
where t is in seconds, find a when t = 0.44 seconds. Round your answer to the nearest hundredth.
A) 9.68 ft
B) -9.68 ft
C) 48.03 ft
D) 28.85 ft
Answer: A
Find the phase shift of the function.
9) y = -3 cos x +
A)
C)
2
2
2
units to the right
B) -3 units up
units to the left
D) -3 units down
Answer: C
10) y = -2 cos (6x + )
A)
2
units to the left
B) 2 units to the right
C) 6 units to the right
D)
Answer: D
Graph the function. Show at least one period.
5
6
units to the left
11) y = 3 sin(2x - )
A)
B)
C)
D)
Answer: C
6
12) y = -2 sin 2x +
2
A)
B)
C)
D)
Answer: D
7
13) y = 2 cos -2x +
2
A)
B)
C)
D)
Answer: A
8
Write the equation of a sine function that has the given characteristics.
14) Amplitude: 2
Period:
7
Phase Shift:
2
A) y = 2 sin 2x +
7
2
B) y = 2 sin (2x - 7)
D) y = 2 sin
C) y = sin (2x + 7)
1
x - 14
2
Answer: B
Solve the problem.
15) A town's average monthly temperature data is represented in the table below:
Average Monthly
Temperature, °F
23.7
26.4
40.1
52.7
69.8
80.4
83.4
78.4
78.4
54.6
37.1
28.6
Month, x
January, 1
February, 2
March, 3
April, 4
May, 5
June, 6
July, 7
August, 8
September, 9
October, 10
November, 11
December, 12
Find a sinusoidal function of the form y = A sin ( x - ) + B that fits the data.
2
x+ 23.7
x+ 29.85
A) y = 83.4 sin
B) y = 53.55 sin
6
3
6
4
C) y = 29.85 sin
6
x-
2
3
+ 53.55
D) y = 23.7 sin
6
x-
4
+ 83.4
Answer: C
16) The number of hours of sunlight in a day can be modeled by a sinusoidal function. In the northern hemisphere,
the longest day of the year occurs at the summer solstice and the shortest day occurs at the winter solstice. In
2000, these dates were June 22 (the 172nd day of the year) and December 21 (the 356th day of the year),
respectively.
A town experiences 11.22 hours of sunlight at the summer solstice and 8.05 hours of sunlight at the winter
solstice. Find a sinusoidal function y = A sin ( x - ) + B that fits the data, where x is the day of the year. (Note:
There are 366 days in the year 2000.)
2
2
+ 8.05
x+ 9.635
A) y = 11.22 sin x B) y = 1.585 sin
3
183
3
C) y = 1.585 sin
183
x-
161
366
+ 9.635
D) y = 11.22 sin
Answer: C
9
172
2
x356
3
+ 9.635
Find the exact value of the expression.
17) tan-1 (-1)
A) -
B)
4
C)
4
7
4
D)
5
4
Answer: A
Use a calculator to find the value of the expression rounded to two decimal places.
5
18) cos-1 8
A) 128.68
B) -0.68
C) -38.68
D) 2.25
C) 2.7
D) 0.3
Answer: D
Find the exact value of the expression. Do not use a calculator.
19) sin [sin-1 (-0.3)]
A) -0.3
B) -2.7
Answer: A
Find the domain of the function f and of its inverse function f-1 .
20) f(x) = 4 tan x + 7
B) Domain of f: ( , )
Domain of f-1 : [3, 11]
A) Domain of f: (
, )
(2k + 1)
Domain of f-1 : x
; k an integer
2
C) Domain of f: x
Domain of f-1 : (
(2k + 1)
; k an integer
2
D) Domain of f: x
(2k + 1)
; k an integer
2
Domain of f-1 : [3, 11]
, )
Answer: C
Find the exact solution of the equation.
21) sin-1 x =
2
A) {-1}
B) {0}
C) { }
D) {1}
Answer: D
22) 6 cos-1 x =
2
A)
2
B)
3
2
C)
B)
3
2
C)
1
2
D)
6
Answer: B
23) 4 cos-1 x =
1
A)
2
Answer: C
10
2
2
D)
4
24) -3 sin-1 (2x) =
3
A)
4
2
4
B)
C) -
1
4
3
4
D) -
Answer: D
Find the exact value of the expression.
1
25) cos sin-1
2
3
2
A)
B)
2
2
C) 0
B)
3
2
C)
B)
3
3
C)
D) 1
Answer: A
2
2
26) sin cos-1 A) -
1
2
2
2
2
2
D) -
Answer: C
27) cos tan-1
A)
3
3
1
2
3
2
D)
3
Answer: D
4
28) cos tan-1 7
A)
7 65
65
B) -
65
7
65
4
C)
D) -
7 65
65
Answer: A
Given that f(x) = sin x, g(x) = cos x, and h(x) = tan x, find the exact value of the composite function.
29) g-1 f
A)
4
B)
4
C) -
4
4
D)
4
Answer: B
Find the exact value of the expression.
3
30) cot-1 3
A) -
6
B)
C)
3
Answer: B
11
6
D) -
3
2 3
3
31) csc-1 A)
B)
6
C) -
3
D) -
6
3
Answer: D
Use a calculator to find the value of the expression in radian measure rounded to two decimal places.
32) sec-1 6
A) 80.41
B) 0.17
C) 1.40
D) 9.59
Answer: C
Write the trigonometric expression as an algebraic expression in u.
33) sin (csc-1 u)
u2 - 1
A) u
B)
u
C)
1
u
D)
u2 + 1
u
D)
1 - u2
Answer: C
34) sec (sin-1 u)
A)
1 - u2
1 - u2
B)
u2 - 1
u
C)
u2 - 1
u2 - 1
Answer: A
Solve the equation on the interval 0
35) tan2 = 3
5 7
11
,
,
,
A)
6 6 6
6
C)
3
<2 .
B)
2 4
5
,
,
3 3
3
,
D)
6
3
,
7
6
,
4
3
Answer: C
36) cot
2
A)
C)
=-
3
3
,
10
9
B)
,
10
9
D)
9
9
9
9
,
10 16
,
9
9
,
10 16 22
,
,
9
9
9
Answer: B
Use a calculator to solve the equation on the interval 0
37) 2 csc = 5
A) {0.41, 2.73}
B) {0.20, 2.94}
< 2 . Round the answer to two decimal places.
C) {0.41}
Answer: A
12
D) {0.20}
Solve the equation on the interval 0
38) cos2 + 2 cos + 1 = 0
A) {2 }
<2 .
B)
4
,
7
4
C)
2
,
3
2
D) { }
Answer: D
39) sec2
A)
- 2 = tan2
B)
4
C) No solution
6
D)
3
Answer: C
Solve the problem.
40) The altitude of a projectile in feet (neglecting air resistance) is given by
16
y = (tan )x x2,
v2 cos2
where x is the horizontal distance covered in feet and v is the initial velocity of the projectile at an angle from
the horizontal. Find the firing angle (in degrees) of a projectile fired at an initial velocity of 100 feet per second
so that it strikes the ground 312.5 feet from the firing point.
A) 45°
B) 22.5°
C) 30°
D) 50°
Answer: A
Use a graphing utility to solve the equation on the interval 0° x < 360°. Express the solution(s) rounded to one decimal
place.
41) sin2 x - 8 sin x - 4 = 0
A) No solution
C) 208.2°, 331.8°
B) 28.2°, 151.8°, 208.2°, 331.8°
D) 28.2°, 151.8°
Answer: C
Find the exact value of the expression.
tan 40° + tan 110°
42)
1 - tan 40° tan 110°
A) - 3
B) -2
C) -
1
2
D) -
3
3
Answer: D
Find the exact value under the given conditions.
7
3
2 21
, < <
; tan = ,
<
43) sin = 25
2
21
2
A)
14 + 24 21
125
B)
<
Find cos ( + ).
-48 - 7 21
125
C)
14 - 24 21
125
D)
48 - 7 21
125
15 - 4 3
16
D)
3 - 15
8
Answer: A
Solve the problem.
44) If sin
A)
=
1
,
4
1-3 5
8
in quadrant II, find the exact value of sin
B)
1+3 5
8
-
3
C)
Answer: B
13
45) If cos
=
1
,
3
in quadrant IV, find the exact value of tan
15 - 4 3
16
A)
3 + 15
8
B)
+
4
15 8
C)
3
D)
1-2 2
1+2 2
Answer: D
Given that f(x) = sin x, g(x) = cos x, and h(x) = tan x, evaluate the given function. The point (x, 3), on the circle
1
x2 + y2 = 4, also lies on the terminal side of an angle in standard position. The point , y , on the circle x 2 + y2 = 1, also
4
lies on the terminal side of an angle
46) f( + )
3 + 15
A)
8
in quadrant IV.
3 - 15
8
B)
C)
-2 2 +
6
3
D)
15 - 4 3
16
Answer: B
47) g( + )
-2 2 +
A)
6
3
B)
1+3 5
8
C)
3 + 15
8
D)
3 - 15
8
3 + 15
1-3 5
D)
3 - 15
1+3 5
Answer: B
48) h( - )
1-3 5
A)
3 + 15
B)
1+3 5
3 - 15
C)
B)
3
3
C) 1
Answer: C
Find the exact value of the expression.
1
3
49) sin cos-1 - sin-1
2
2
A) 0
D)
2 3
2
Answer: A
Use the information given about the angle , 0
2 , to find the exact value of the indicated trigonometric function.
24
,
<2 <
Find sin .
50) cos 2 = 25 2
A)
7
5
7
5
C) -
7 2
10
D)
24
25
C) -
24
25
D) -
B) -
7 2
10
Answer: D
Find the exact value of the expression.
3
51) sin 2 cos-1 5
A)
12
25
B)
Answer: C
14
12
25
52) cos sin-1
A)
2
1
+ 2 sin-1 3
3
7 5+8 2
27
B)
2 3
5
C)
2 6
5
D)
2 3 + 2 10
9
Answer: A
Use the information given about the angle , 0
2 , to find the exact value of the indicated trigonometric function.
1
Find cos .
53) sin = , tan > 0
4
2
6
4
A)
8 - 2 15
4
B)
8 + 2 15
4
C)
D)
10
4
Answer: C
54) cos
=-
3
,
5
<
<
3
2
5
5
A)
Find cos
B) -
2
.
5
5
C) -
30
10
D)
30
10
C) -
5
5
D)
10
10
D)
6
4
Answer: B
55) sin
=-
3
,
5
2
<
<
5
5
A)
Find sin
B) -
2
.
30
10
Answer: D
56) cos(2 ) =
1
, 0<
4
8 - 2 10
4
A)
<
2
Find sin .
10
4
B)
10 - 2 6
4
C)
Answer: D
Use the Half-angle Formulas to find the exact value of the trigonometric function.
57) cos 22.5°
1
1
1
2+ 2
2+ 2
2- 2
A) B)
C) 2
2
2
D)
1
2
2-
2
D)
1
2
2+
3
Answer: B
58) sin
12
A)
1
2
1-
3
B)
1
2
2-
3
C)
Answer: B
15
1
2
1-
3
59) sin
8
A) -
1
2
2-
3
B)
1
2
2-
2
C)
Answer: B
16
1
2
1-
2
D) -
1
2
2-
2