Download Math2412-PreCalculus-TestReview2

Math2412-PreCalculus-TestReview2-Spring2016
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Complete the identity.
1
=?
1) sec x sec x
A) sin x tan x
2)
3) 2 tan x - (1 + tan x)2 = ?
A) 1
D) 1 + cot x
2)
B) sec x csc x
C) sin x tan x
D) -2 tan2 x
3)
B) - sec2 x
C) 0
D) 1 - sin x
csc x cot x
=?
sec x
A) cot2 x
5)
C) -2 tan2 x
B) sec x csc x
sin x cos x
+
=?
cos x sin x
A) 1 + cot x
4)
1)
4)
B) csc2 x
D) sec2 x
C) 1
cos x + sin x sin x - cos x
=?
cos x
sin x
A) 2 - sec x csc x
5)
B) 2 + sec x csc x
C) 1 - sec x csc x
D) sec x csc x
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Verify the identity.
6) cot 2 x + csc 2 x = 2 csc 2 x - 1
6)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the exact value of the expression.
7) cos
A)
9
1
2
-
7)
18
B)
1
4
C) 1
D)
3
2
Identify and in the following expression which is the right side of the formula for cos ( - ).
8) cos (155°) cos (35°) + sin (155°) sin (35°)
A) = 35°, = 155°
B) = 155°, = 35°
C) = - 35°, = 155°
D) = -155°, = 35°
Find the exact value of the expression.
9) cos (165°) cos (45°) + sin (165°) sin (45°)
1
A) B) -2
2
8)
9)
C) -
1
3
2
D) -
3
Complete the identity.
5
=?
10) cos x 6
A)
C)
10)
3
(cos x - sin x)
2
1
(- 3 cos x + sin x)
2
B) -
3
(cos x + sin x)
2
D) -
3
(cos x - sin x)
2
Use the given information to find the exact value of the expression.
4
2
11) sin = , lies in quadrant II, and cos = , lies in quadrant I
5
5
A)
8 + 3 21
25
B)
-6 + 4 21
25
C)
Find cos ( - ).
6 - 4 21
25
D)
8 - 3 21
25
Find the exact value by using a sum or difference identity.
12) sin 165°
A) - 2( 3 + 1)
11)
12)
B) - 2( 3 - 1)
2( 3 - 1)
4
C) -
Find the exact value of the expression.
13) cos 20° cos 40° - sin 20° sin 40°
3
1
A)
B)
4
2
2( 3 - 1)
4
D)
13)
3
C)
D)
1
2
Complete the identity.
14) sin x +
2
A) sin x
=?
14)
B) -cos x
C) -sin x
D) cos x
Use the given information to find the exact value of the expression.
5
3
, lies in quadrant III, and cos = - , lies in quadrant II
15) tan =
12
5
A)
56
65
B)
63
65
C)
Find the exact value by using a difference identity.
16) tan 105°
2+ 3
A) 2 + 3
B)
4
16
65
Find sin ( + ).
D) -
33
65
16)
C) -2 -
3
2- 3
D)
4
Use trigonometric identities to find the exact value.
tan 155° - tan 35°
17)
1 + tan 155° tan 35°
A) -2
B) -
15)
17)
1
2
C) -
2
3
3
D) - 3
Use the given information to find the exact value of the expression.
8
, lies in quadrant II
Find tan 2 .
18) sin =
17
A)
240
289
B) -
240
161
C)
240
161
18)
D) -
159
161
Use the figure to find the exact value of the trigonometric function.
19) Find sin 2 .
5
4
8
A) 25
19)
3
B)
7
25
C) -
7
25
D)
24
25
20) Find cos 2 .
20)
5
4
A) -
1
5
3
B)
7
25
C) -
7
25
D)
24
25
Write the expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression.
21) cos2 30° - sin2 30°
21)
A) -
3
2
B)
1
2
C) -
1
2
D)
3
2
Rewrite the expression as an equivalent expression that does not contain powers of trigonometric functions greater
than 1.
22) sin4 x
22)
A)
3 5
+ cos 2x
8 8
B)
3
1
- 2 cos 2x + cos 4x
2
2
C)
3 1
1
- cos 2x + cos 4x
8 2
8
D)
3 3
- cos 2x
2 2
23) cos3 x
3
1
A) cos x + cos 3x
4
4
C)
3
1
B) cos x - cos 3x
4
4
3
1
cos x + cos 3x + cos 2x
4
4
D)
3
3
1
cos x - cos 3x - cos 2x
4
4
23)
Express the product as a sum or difference.
24) sin 6x sin 4x
1
A) (cos 10x - cos 2x)
2
C)
24)
1
B) (cos 2x - cos 10x)
2
1
(sin 10x + cos 2x)
2
D) sin2 24x2
25) sin 7x cos 3x
1
A) (sin 10x + sin 4x)
2
C) sin (cos 21x 2 )
B)
1
(cos 10x - cos 4x)
2
D)
1
(sin 10x + cos 4x)
2
25)
Use the given information to find the exact value of the trigonometric function.
12
, lies in quadrant IV
Find cos .
26) sin = 13
2
A) -
3 13
13
B)
5
26
C) -
Express the product as a sum or difference.
27) cos 5x cos 8x
1
(cos 3x + cos 13x)
2
D)
Express the sum or difference as a product.
28) sin 8x + sin 4x
A) 2 cos 6x sin 2x
B) 2 sin 12x
29) cos 9x - cos 3x
A) -2 cos 6x sin 3x
2 13
13
D)
3 13
13
27)
1
B) (cos 13x - sin 3x)
2
A) cos2 28x 2
C)
26)
B) -2 sin 6x sin 3x
1
(cos 13x - cos 3x)
2
C) 2 sin 6x sin 2x
D) 2 sin 6x cos 2x
C) 2 cos 6x cos 3x
D) 2 cos 3x
28)
29)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Verify the identity.
cos x + cos y
x- y
= cot
30)
sin x - sin y
2
30)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find all solutions of the equation.
31) 2 cos x - 1 = 0
A) x =
C) x =
3
3
+ 2n or x =
+ n or x =
31)
3
6
+ 2n
B) x =
+n
D) x =
4
3
3
+ n or x =
3
+ 2n or x =
+n
6
+ 2n
Solve the equation on the interval [0, 2 ).
2
32) cos 2x =
2
A)
32)
7
9
15
,
,
8 8
8
8
C) 0,
B)
2
4
, ,
3
3
D) no solution
33) 2 sin2 x = sin x
5
A) ,
6 6
B)
3
,
2
3
C)
,
C) 0, ,
34) sin2 x - cos2 x = 0
A)
3
5
7
,
,
4 4
4
4
,
6
,
5
6
D)
2
,
3
2
, ,
2 3 3
34)
3
5
7
,
,
B) ,
4 4
4
4
4
,
4 3
D)
,
4 6
Solve the equation on the interval [0, 2 ).
35) (tan x + 1) (cos x + 1) = 0
A)
4
, ,
B) 0,
4
33)
35)
4
,
C) 0,
4
4
,
4
D)
Solve the triangle.
36)
4
,
4
,
36)
80°
7
55°
A) B = 40°, a = 9.75, c = 8.11
C) B = 50°, a = 8.11, c = 9.75
B) B = 45°, a = 8.11, c = 9.75
D) B = 45°, a = 9.75, c = 8.11
Two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one
triangle, two triangles, or no triangle at all. Solve each triangle that results. Round lengths to the nearest tenth and
angle measures to the nearest degree.
37) B = 80°, b = 5, a = 23
37)
A) A = 39°, C = 60°, c = 30
B) A = 40°, C = 60°, c = 28
C) no triangle
D) A = 41°, C = 60°, c = 32
38) C = 35°, a = 18.7, c = 16.1
A) A1 = 42°, B1 = 103°, b1 = 27.4;
B) no triangle
A2 = 138°, B2 = 7°, b2 = 3.4
C) A1 = 103°, B1 = 42°, b1 = 27.4;
D) A = 42°, B = 103°, b = 27.4
A2 = 7°, B2 = 138°, b2 = 3.4
5
38)
Find the area of the triangle having the given measurements. Round to the nearest square unit.
39) A = 27°, b = 14 inches, c = 5 inches
A) 14 square inches
B) 31 square inches
C) 16 square inches
D) 33 square inches
Solve the triangle. Round lengths to the nearest tenth and angle measures to the nearest degree.
40)
7
39)
40)
6
8
A) A = 47°, B = 58°, C = 75°
C) A = 58°, B = 75°, C = 47°
B) A = 47°, B = 75°, C = 58°
D) A = 58°, B = 47°, C = 75°
41) a = 6, b = 8, C = 106°
A) c = 14.1, A = 33°, B = 41°
C) c = 17, A = 29°, B = 45°
B) c = 11.2, A = 31°, B = 43°
D) no triangle
Use Heron's formula to find the area of the triangle. Round to the nearest square unit.
42) a = 17 yards, b = 17 yards, c = 17 yards
A) 125 square yards
B) 134 square yards
C) 128 square yards
D) 131 square yards
Use a polar coordinate system to plot the point with the given polar coordinates.
-5
43) -4,
4
6
41)
42)
43)
A)
B)
C)
D)
Find another representation, (r, ), for the point under the given conditions.
44) 2,
2
, r < 0 and 0 <
A) -2,
5
2
<2
44)
B) -2, -
3
2
C) -2, -
1
2
Polar coordinates of a point are given. Find the rectangular coordinates of the point.
45) (4, -180°)
A) (0, -4)
B) (0, 4)
C) (4, 0)
46) 7,
D) -2,
3
2
D) (-4, 0)
2
3
A) -
45)
46)
7 -7 3
,
2
2
B)
7 -7 3
,
2
2
C)
7 7 3
,
2
2
D) -
7 7 3
,
2
2
The rectangular coordinates of a point are given. Find polar coordinates of the point. Express in radians.
47) (7, -7)
7
7
7
A) 7 2, B) 7 2,
C) 7, D) 7,
4
4
4
4
7
47)
48) (-5, 0)
48)
3
B) 5,
2
A) (5, 0)
C) (5, )
D) 5,
2
Convert the rectangular equation to a polar equation that expresses r in terms of .
49) (x - 13)2 + y2 = 169
A) r = -26 sin
+ 169
C) r = 26 cos
D) r = 26 sin
Convert the polar equation to a rectangular equation.
50) r cos = 7
A) x2 + y2 = 7
B) x = 7
51) r2 sin 2 = 9
A) xy = 9
49)
B) r2 = 26 cos
C) y2 = 7
50)
D) y = 7
51)
9
B) xy =
2
C) y2 = 9
8
D) x2 + y2 = 9
Graph the polar equation.
52) r = 1 - sin
52)
A)
B)
C)
D)
Write the complex number in polar form. Express the argument in radians.
53) 2 - 2i
5
5
7
7
+ i sin
+ i sin
A) 2 cos
B) 2 2 cos
4
4
4
4
C) 2 cos
7
7
+ i sin
4
4
D) 2 2 cos
9
5
5
+ i sin
4
4
53)
Find the product of the complex numbers. Leave answer in polar form.
54) z1 = 4i
54)
z2 = -6 + 6i
A) 24 2 sin
5
5
+ i cos
4
4
B) 24 2 cos
3
3
+ i sin
8
8
C) 24 2 sin
3
3
+ i cos
8
8
D) 24 2 cos
5
5
+ i sin
4
4
Find the quotient
55) z1 =
z2 =
z1
z2
of the complex numbers. Leave answer in polar form.
1
2
2
cos
+ i sin
5
3
3
55)
1
cos
+ i sin
4
4
4
A)
5
5
5
cos + i sin 4
12
12
B)
4
5
5
cos
+ i sin
5
12
12
C)
4
8
8
cos + i sin
5
3
3
D)
1
11
11
cos
+ i sin
20
12
12
Find the unit vector that has the same direction as the vector v.
56) v = 3i - 4j
5
5
3
4
A) u = 15i - 20j
B) u = i - j
C) u = i - j
3
4
5
5
Write the vector v in terms of i and j whose magnitude v
57) v = 10, = 120°
A) v = -5 2i + 5 2j
C) v = 5 3i - 5j
and direction angle
4
3
D) u = i - j
5
5
are given.
56)
57)
B) v = -5i + 5 3j
D) v = 5i - 5 3j
Use the given vectors to find the specified scalar.
58) u = 6i + 10j, v = -2i - 8j, w = -4i - 2j; Find u · (v + w).
A) -136
B) -120
C) -44
D) -92
Find the angle between the given vectors. Round to the nearest tenth of a degree.
59) u = 3j, v = 8i + 2j
A) 14°
B) 68.7°
C) 43.3°
D) 76°
Decompose v into two vectors v1 and v2 , where v1 is parallel to w and v2 is orthogonal to w.
60) v = i + 9j, w = i + j
11
9
7
(i + j), v2 = - i + j
A) v1 =
B) v1 = 10(i + j), v2 = -8i + 8j
2
2
2
C) v1 = 5(i + j), v2 = -4i + 4j
D) v1 = 5(i + j), v2 = 4i - 4j
10
58)
59)
60)
Answer Key
Testname: UNTITLED1
1)
2)
3)
4)
5)
A
B
B
A
D
cot 2 x + csc 2 x = csc 2 x - 1 + csc 2 x = 2 csc 2 x - 1.
D
B
A
C
B
D
D
B
D
C
D
B
D
C
B
C
A
B
A
A
C
D
B
x+ y
x- y
x- y
2 cos
cos
cos
2
2
2
cos x + cos y
x- y
=
=
= cot
30)
sin x - sin y
x- y
x+ y
x- y
2
2 sin
cos
sin
2
2
2
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
23)
24)
25)
26)
27)
28)
29)
31)
32)
33)
34)
35)
36)
37)
38)
39)
40)
41)
42)
43)
44)
45)
46)
A
A
C
B
A
B
C
A
C
D
B
A
C
D
D
D
11
Answer Key
Testname: UNTITLED1
47)
48)
49)
50)
51)
52)
53)
54)
55)
56)
57)
58)
59)
60)
B
C
C
B
B
B
B
D
B
C
B
A
D
C
12