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Transcript 
Name: ________________________ Class: ___________________ Date: __________
Practice Test3 (Trigonometry)
ID: A
Instructor: Koshal Dahal
Multiple Choice Questions
SHOW ALL WORK, EVEN FOR MULTIPLE CHOICE QUESTIONS, TO RECEIVE CREDIT.
____
____
____
1. Find the values of the trigonometric functions of t if sec t 
0
0
1 0
1
, cost  , tan t  
, cot t  
1
1
0
1
a.
sint  
b.
sin t 
0
1 0
1
0
, cost  , tan t  , cot t 
1
0
1
1
c.
sint 
0
0
1 0
0
, cost  , tan t 
, cot t 
1
1
0
1
2. Find the values of the trigonometric functions of t if tant  
a.
sint  
b.
sint 
c.
sint  
11
11
5 11
5
, cost  , tan t  
, cot t 
6
5
11
6
11
11
5 11
5
, cost  , tan t  
, cot t 
6
5
11
6
11
11
5 11
5
, cost  , tan t  
, cot t  
6
5
11
6
3. Rewrite the expression as an algebraic expression in x.
tan (sin – 1 x)
1
a.
x2  1
x
b.
1  x2
____
1
, and the terminal point of t is in quadrant IV.
1
c.
1  x2
d.
x2  1
4. Rewrite the expression as an algebraic expression in x.
sin (cos – 1 x)
a.
b.
x2  1
x–1
c.
d.
e.
1  x2
1–x
x
1
11
and cos t  0.
5
Name: ________________________
____
ID: A
5. Find the exact value of the expression.
ÊÁ
ˆ
ÁÁ 1 3 ˜˜˜
Á
˜˜
cos ÁÁ sin
ÁÁ
2 ˜˜˜
Ë
¯
a.
b.
____

2
1
2
c.
2
2
d.
3
2
6. Simplify the expression.
sin 14x
sin13x  sin x
sin6x
a.
sin7x
cos 6x
b.
cos 7x
sin 13x
c.
sin6x
sin7x
d.
sin6x
cos 7x
e.
cos 6x
____
7. Find all solutions of the equation.
2 cos x 
2 0
Select the correct answer, where k is any integer:
4

 2k  ,
 2k 
a.
5
5
7

b.
 2k  ,
 2k 
4
4
7

c.
 k ,
 k
4
4
9

d.
 2k  ,
 2k 
5
5
2
Name: ________________________
____
ID: A
8. Find all solutions of the equation.
2 sin x  1  0
Select the correct answer, where k is any integer:
11

 k ,
 k
a.
6
6
5

b.
 k ,
 k
6
6
5

c.
 2k  ,
 2k 
6
6
11

d.
 2k  ,
 2k 
6
6
____
9. Find all solutions of the following equation.
4 cos 2 x  3  0
Select the correct answer, where k is any integer:
5
7
11

 2k  ,
 2k  ,
 2k  ,
 2k 
a.
6
6
6
6
11

b.
 k ,
 k
6
6
5
7
11

c.
 k ,
 k ,
 k ,
 k
6
6
6
6
5

d.
 2k  ,
 2k 
6
6
____ 10. Find all solutions of the following equation.
sec 2 x  4  0
Select the correct answer, where k is any integer:

2
 2k  ,
 2k 
a.
3
3

5
b.
 k ,
 k
3
3
2
4
5

c.
 2k  ,
 2k  ,
 2k  ,
 2k 
3
3
3
3
2
4
5

d.
 k ,
 k ,
 k ,
 k
3
3
3
3
3
Name: ________________________
ID: A
____ 11. Find all solutions of the following equation.
4 cos 2 x – 4 cos x + 1 = 0
Select the correct answer, where k is any integer:
7

a.
 2k  ,
 2k 
4
4
5

b.
 2k  ,
 2k 
3
3
5

c.
 2k  ,
 2k 
6
6
3

d.
 2k  ,
 2k 
4
4
____ 12. Find all solutions of the following equation.
sin 2 x = –3 sin x + 4
Select the correct answer, where k is any integer:
a.
b.
c.
d.

2

4
k

2
 2k 
 2k  ,
3
 2k 
4
 k
____ 13. Find all solutions of the following equation.
Ê
cos x ÁÁÁ 2sinx 
Ë
ˆ
2 ˜˜˜  0
¯
Select the correct answer, where k is any integer:

5
 2k  ,
 2k 
a.
2
4

3
b.
 2k  ,
 2k 
2
2
5
7
c.
 2k  ,
 2k 
4
4

3
5
7
d.
 2k  ,
 2k  ,
 2k  ,
 2k 
2
2
4
4
4
Name: ________________________
ID: A
____ 14. Find all solutions of the following equation.
sin x cos x – 2 sin x = 0
Select the correct answer, where k is any integer:

3
a.
 2k  ,
 2k 
4
4
b. 2k  ,   2k 
c.
d.


 k
6
3

 2k  ,
 2k 
2
2
2
 k ,
____ 15. Find all solutions of the following equation in the interval [0,2 ).
tan x – 3 cot x = 0
Select the correct answer, where k is any integer:
 2
,
a.
3 3
 2 4 5
b.
,
,
,
3 3
3
3
2

c.
 2k  ,
 2k 
3
3
2
4
5

d.
 2k  ,
 2k  ,
 2k  ,
 2k 
3
3
3
3
____ 16. Use an addition or subtraction formula to simplify the following equation. Then find all the solutions in the
ÍÈÍ  ˆ˜
interval ÍÍÍÍ 0, ˜˜˜˜ .
ÍÎ 4 ¯
cos x cos 7 x – sin x sin 7 x = 0
a.
b.
c.
d.

16

8
,
3
8
,
3
16

8

16
5
Name: ________________________
ID: A
____ 17. Use an addition or subtraction formula to simplify the following equation. Then find all the solutions in the
È
interval ÍÍÍÎ 0,2 ˆ˜¯ .
sin 4 x cos 3 x – cos 4 x sin 3 x = 0
a. 
b. 0,  , 2
c. 0, 2
d. 0, 
ÊÁ 
____ 18. Plot the point that has the polar coordinates ÁÁÁÁ 5,
Ë 4
a.
d.
b.
e.
c.
6
ˆ˜
˜˜ .
˜˜
¯
Name: ________________________
ID: A
ÁÊ
7
____ 19. Plot the point that has the polar coordinates ÁÁÁÁ 3,
6
Ë
a.
d.
b.
e.
c.
7
˜ˆ˜
˜˜ .
˜
¯
Name: ________________________
ID: A
____ 20. Determine which point in the figure, P, Q, R, or S, has the given polar coordinates.
ÊÁ
ÁÁ 4, 25
ÁÁ
4
Ë
a.
b.
c.
d.
ˆ˜
˜˜
˜˜
¯
P
S
Q
R
8
Name: ________________________
ID: A
____ 21. Determine which point in the figure, P, Q, R, or S, has the given polar coordinates.
ÊÁ
ÁÁ 4, 21
ÁÁ
4
Ë
a.
b.
c.
d.
ˆ˜
˜˜
˜˜
¯
P
Q
S
R
9
Name: ________________________
ID: A
____ 22. Determine which point in the figure, P, Q, R, or S, has the given polar coordinates.
ÊÁ
ÁÁ 4, 25
ÁÁ
4
Ë
a.
b.
c.
d.
ˆ˜
˜˜
˜˜
¯
P
R
S
Q
____ 23. Find the modulus of the complex number.
–9
a.
b.
c.
d.
9
0
–9
10
____ 24. Find the modulus of the complex number.
1 
a.
b.
c.
d.
3
i
3
5 3
2
9
2 3
3
2
3
10
Name: ________________________
ID: A
____ 25. Sketch the complex number z, also sketch 2z, –z and
z  1 
2i
a.
d.
b.
e.
c.
11
1
z on the same complex plane.
2
Name: ________________________
ID: A
____ 26. Sketch the complex number z and its complex conjugate z on the same complex plane.
z  7  4i
a.
d.
b.
e.
c.
12
Name: ________________________
ID: A
____ 27. Sketch z 1 ,z 2 ,z 1  z 2 , and z 1 z 2 on the same complex plane.
z 1  2  i, z 2  2  3i
a.
d.
b.
e.
c.
13
Name: ________________________
ID: A
____ 28. Write the complex number 4  4 3i in polar form with argument  between  and  .
ÁÊ  ˜ˆ
ÁÊ  ˜ˆ
a. 8 cos ÁÁÁÁ ˜˜˜˜  8i sin ÁÁÁÁ ˜˜˜˜
Ë3¯
Ë3¯
ÊÁ  ˆ˜
ÊÁ  ˆ˜
b. 4 cos ÁÁÁÁ ˜˜˜˜  4i sin ÁÁÁÁ ˜˜˜˜
Ë3¯
Ë3¯
ÊÁ  ˆ˜
ÊÁ  ˆ˜
c. cos ÁÁÁ ˜˜˜  i sin ÁÁÁ ˜˜˜
ÁË 3 ˜¯
ÁË 3 ˜¯
____ 29. Write the complex number 4 + 4 i in polar form with argument  between 0 and 2 .

 4i cos
a.
4 sin
b.
cos
c.
4 2 cos
4

 i sin
4

4

4

4
 4 2 i sin

4
____ 30. Write the complex number 5 i in polar form with argument  between 0 and 2 .
a.
5 cos
b.
5 cos
c.

3

4

5 cos
2
 5i sin
 5i sin
 5i sin

3

4

2
____ 31. Write the complex number –2 in polar form with argument  between 0 and 2 .

 2i sin

a.
2 cos
b.
2 cos   2i sin 
c.
2 cos
4

2
 2i sin
4

2
____ 32. Write the complex number i( 8 – 8 i ) in polar form with argument  between 0 and 2 .
a.
8 2 cos
b.
cos
c.
8 sin

4

4

4
 8 2 i sin
 i sin

4
 8i cos

4

4
14
Name: ________________________
____ 33. Find the quotient
z 1  cos
z 2  cos

5
 i sin

ID: A
z1
, in polar form, if:
z2

5

 i sin
6
6
11
11
cos
 i sin
30
30
a.
b.
cos
c.
cos

11

30
 i sin
 i sin

11

30
____ 34. Find the product z 1 z 2, expressed in polar form, if:
z 1  cos
z 2  cos

2

12
 i sin
 i sin


2

12

 i sin
12
12
5
5
cos
 i sin
12
12
7
7
cos
 i sin
12
12
cos
a.
b.
c.
____ 35. Find the quotient
z1
, in polar form, if:
z2
ˆ
Ê
7 ÁÁ cos 75   i sin 75  ˜˜
¯
Ë
ˆ
Ê

˜
Á
 3 7 Á cos 60  i sin 60 ˜
¯
Ë
7 ÊÁ

ˆ
Á cos 75  i sin 60 ˜˜
3Ë
¯
1 ÊÁ
ˆ
Á cos 15   i sin 15  ˜˜
3Ë
¯
ÊÁ

ˆ
3 Á cos 15  i sin 15 ˜˜
¯
Ë
z1 
z2
a.
b.
c.
15
Name: ________________________
ID: A
____ 36. Write z 1 and z 2 in polar form, and then find the product z 1 z 2 if:
Ê
ˆ
z 1  2 ÁÁÁ 3  i ˜˜˜
Ë
¯
ÊÁ
ˆ
z 2  3 ÁÁ 1  i 3 ˜˜˜
Ë
¯
Express your answer in polar form.
ÊÁ

 ˆ˜
a. 3 ÁÁÁÁ cos  i sin ˜˜˜˜
4
4¯
Ë
ÊÁ

 ˆ˜
b. 24 ÁÁÁ cos  i sin ˜˜˜
ÁË
2
2 ˜¯
ÊÁ

 ˆ˜
c. 2 ÁÁÁÁ cos  i sin ˜˜˜˜
2
2¯
Ë
____ 37. Write z 1 and z 2 in polar form, and then find the quotient
Ê
ˆ
z 1  ÁÁÁ 2 5  2i 5 ˜˜˜
Ë
¯
z 2  (2  2i)
a.
b.
c.
5i
2i
5
____ 38. Find the indicated power using DeMoivre's Theorem.
( 1 + i ) 16
a. –256
b. 256
c. –256 i
____ 39. Find the indicated power using DeMoivre's Theorem.
( 1 – i ) 12
a. 64 i
b. –64
c. 64
____ 40. Find the indicated power using DeMoivre's Theorem.
ÊÁ
ÁÁ 2
2
ÁÁ
ÁÁ 2  2
Á
Ë
a. i
b. 1
c. –1
ˆ˜ 12
˜˜
i ˜˜˜
˜˜
¯
16
z1
if:
z2
Name: ________________________
ID: A
____ 41. Find the indicated power using DeMoivre's Theorem.
ˆ 10
ÁÊÁ 3
3 ˜˜˜˜
ÁÁ
i˜
ÁÁ 
ÁÁ 2
2 ˜˜˜
Ë
¯
ÊÁ
ˆ
3 ˜˜˜˜
ÁÁÁ 1
i˜
a. 243 ÁÁ 
ÁÁ 2
2 ˜˜˜
Ë
¯
ÊÁ
ˆ
ÁÁ 3 1 ˜˜˜
Á
 i ˜˜˜
b. 243 ÁÁ
2 ˜˜
ÁÁ 2
Ë
¯
ÁÊÁ 3 1 ˜ˆ˜
Á
˜
 i ˜˜˜
c. 243 ÁÁÁ
ÁÁ 2
2 ˜˜
Ë
¯
____ 42. Find the cube roots of:
125 i
a.
b.
c.
ÊÁ

 ˆ˜ ÊÁ
3
5
5 ˆ˜˜˜ ÊÁÁÁ
3 ˆ˜˜˜
,
5
cos
5 ÁÁÁ cos  i sin ˜˜˜˜ , 5 ÁÁÁÁ cos
 i sin

i
sin
˜
Á
ÁË
6
6¯ Ë
6
6 ˜¯ ÁË
2
2 ˜˜¯
ÊÁ
ÊÁ
ÊÁ

 ˆ˜
5
5 ˆ˜˜˜
ÁÁ cos 3  i sin 3
,
5
5 ÁÁÁÁ cos  i sin ˜˜˜˜ , 5 ÁÁÁÁ cos
 i sin
˜
ÁÁ
˜
6
6
6
6
2
2
Ë
¯
Ë
¯
Ë
ÊÁ

 ˆ˜ ÊÁ
13
7
7 ˆ˜˜˜ ÊÁÁÁ
13 ˆ˜˜˜
,
5
cos
5 ÁÁÁÁ cos  i sin ˜˜˜˜ , 5 ÁÁÁÁ cos
 i sin

i
sin
˜
Á
6
6¯ Ë
6
6 ˜¯ ÁË
6
6 ˜˜¯
Ë
ˆ˜
˜˜
˜˜
¯
Short Answer
43. Use an appropriate half-angle formula to find the exact value of the expression.
sin15
44. Use an appropriate half-angle formula to find the exact value of the expression.
tan22.5
45. Use an appropriate half-angle formula to find the exact value of the expression.
sin

12
17
Name: ________________________
ID: A
46. Simplify each expression by using a double-angle formula.
(a.)
2 tan2
= tan ÊÁË _ _ _ _ _ _ _ _ _ _  ˆ˜¯
1  tan 2 2
(b.)
2tan2
= __________
1  tan 2 2
47. Simplify each expression by using a half-angle formula.
(a.)
1  cos 30
sin ÊÁË _ _ _ _ _ _ _ _ _ _  ˆ˜¯
2
(b.)
1  cos 10
= __________
2
48. Write the product as a sum.
sin x sin5x
49. Write the product as a sum.
7cos 6x cos 7x
50. Write the sum as a product.
sin2x  sin7x
51. Write the sum as a product.
sin3x  sin4x
52. Find the value of the product.
3 sin 37.5 cos 7.5
53. Simplify the expression.
2tanx
1  tan 2 x
54. Find all solutions of the equation.
2 cos x  1  0
18
Name: ________________________
ID: A
55. Find all solutions of the equation.
tan 9 x  81 tan x  0
56. Find all solutions of the equation.
4 cos 2 x  4 cos x  1  0
57. Find all solutions of the equation.
4 sin x cos x  2 sin x  2 cos x  1  0
58. Consider the equation.
2 tan x  19
(a) Find all solutions of the equation.
x = __________
(b) Use a calculator to solve the equation in the interval
ÍÈÍ 0,2 ˆ˜ , correct to five decimal places.
ÍÎ
¯
x = __________
59. Use an addition or subtraction formula to simplify the equation. Then find all solutions in the interval
ÈÍ
ÍÍ 0,2 ˆ˜ .
Î
¯
sin4x cos x  cos 4x sin x 
3
2
60. Solve the equation by first using a sum-to-product formula.
sinx  sin 3x  0
19
Name: ________________________
ID: A
61. If a projectile is fired with velocity v 0 at an angle  , the its range, the horizontal distance it travels (in feet), is
modeled by the function
R ( ) 
v 20 sin2
32
.
If v 0  2,000 ft/s, what angle (in degrees) should be chosen for the projectile to hit a target on the ground
5000 ft away? Please give the answer to five decimal places.
  __________ 
62. A point is graphed in rectangular form. Find polar coordinates for the point, with r  0 and 0    2 .
63. A point is graphed in polar form. Find its rectangular coordinates.
20
Name: ________________________
ID: A
ÁÊ
7
64. Find the rectangular coordinates for the point whose polar coordinates are ÁÁÁÁ 1,
2
Ë
Ê
65. Convert the rectangular coordinates ÁÁÁ 0,
Ë
ˆ
7 ˜˜˜ to polar coordinates with r  0 and 0    2 .
¯
66. Convert the equation to polar form.
x2
67. Convert the polar equation to rectangular coordinates.
  7
68. Convert the polar equation to rectangular coordinates.
r
2
1  5sin 
69. Write the complex numbers in polar form with argument  between 0 and 2 .
(a) 1 
(b)
3 i  __________
2
2i  __________
70. Find the product z 1 z 2 and the quotient
ÊÁ
2
2
z 1  7 ÁÁÁ cos
 i sin
ÁË
3
3
z1
. Express your answer in polar form.
z2
ˆ˜
Ê
ˆ
˜˜ , z  2 ÁÁÁ cos   i sin  ˜˜˜
˜˜ 2
ÁÁ
3
3 ˜˜¯
¯
Ë
(a) z 1 z 2  __________
(b)
˜ˆ˜
˜˜ .
˜
¯
z1
= __________
z2
71. Find the indicated power using DeMoivre's Theorem.
ÊÁ
ˆ4
ÁÁ 2 3  2i ˜˜˜
Ë
¯
72. Find the indicated power using DeMoivre's Theorem.
(1  i) 8
21
ID: A
Practice Test3 (Trigonometry)
Answer Section
Instructor: Koshal Dahal
MULTIPLE CHOICE
1.
2.
3.
4.
5.
6.
7.
8.
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12.
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15.
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32.
33.
34.
35.
36.
37.
38.
39.
40.
ANS:
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ANS:
A
C
B
C
B
E
B
C
A
C
B
A
D
B
B
D
D
A
C
D
D
C
A
C
B
D
D
A
C
C
B
A
C
C
B
B
C
B
B
C
PTS:
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PTS:
PTS:
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
ID: A
41. ANS: A
42. ANS: A
PTS: 1
PTS: 1
SHORT ANSWER
43. ANS:
2
2
3
PTS: 1
44. ANS:
2 1
PTS: 1
45. ANS:
1 ÁÊ
Á 6 
4 ÁË
ˆ
2 ˜˜˜
¯
PTS: 1
46. ANS:
4; tan(4)
PTS: 1
47. ANS:
15; sin(5)
PTS: 1
48. ANS:
1 Ê
 Á cos(4x)  cos(6x) ˆ˜¯
2 Ë
PTS: 1
49. ANS:
7 Ê
 Á cos(13x)  cos(x) ˆ˜¯
2 Ë
PTS: 1
50. ANS:
ÁÊ 9x ˜ˆ
ÁÊ 5x ˜ˆ
2 cos ÁÁÁÁ ˜˜˜˜  sin ÁÁÁÁ ˜˜˜˜
Ë 2 ¯
Ë 2 ¯
PTS: 1
2
ID: A
51. ANS:
ÊÁ 7x ˆ˜
ÊÁ x ˆ˜
2 sin ÁÁÁÁ ˜˜˜˜  cos ÁÁÁÁ ˜˜˜˜
Ë 2 ¯
Ë2¯
PTS: 1
52. ANS:
ˆ
3 ÊÁ
 ÁÁ 2  1 ˜˜˜
4 Ë
¯
PTS: 1
53. ANS:
sin(2x)
PTS: 1
54. ANS:
2
4
x
 2k   ,
 2k  
3
3
PTS: 1
55. ANS:
k 
x
3
PTS: 1
56. ANS:
x

3
 2k   ,
5
 2k  
3
PTS: 1
57. ANS:
11
2
4
7
 2k   ,
 2k   ,
 2k   ,
 2k  
6
6
3
3
PTS: 1
58. ANS:
1.46592  k   ; 1.46592, 4.60751
PTS: 1
59. ANS:
 2 7 8 13 14 19 4 5 26
,
,
,
,
,
,
,
,
,
15 15 15 15 15
15
15
3
3
15
PTS: 1
60. ANS:
1
x  k 
2
PTS: 1
3
ID: A
61. ANS:
1.14622, 88.85378
PTS: 1
62. ANS:
ÊÁ
ÁÁ 2 2, 
ÁÁ
4
Ë
ˆ˜
˜˜
˜˜
¯
PTS: 1
63. ANS:
ÊÁ
ˆ
ÁÁ 5 3 5 ˜˜˜
ÁÁ
˜
ÁÁ 2 , 2 ˜˜˜
Á
˜
Ë
¯
PTS: 1
64. ANS:
(0, 1)
PTS: 1
65. ANS:
ÊÁ
ÁÁ 7 , 3
ÁÁ
2
Ë
ˆ˜
˜˜
˜˜
¯
PTS: 1
66. ANS:
r  2sec ( )
PTS: 1
67. ANS:
y0
PTS: 1
68. ANS:
x 2  24y 2  20y  4  0
PTS: 1
69. ANS:
ÁÊÁ ÊÁ 5
2 ÁÁÁ cos ÁÁÁÁ
ÁË Ë 3
ˆ˜
Ê
˜˜  i  sin ÁÁÁ 5
˜˜
ÁÁ 3
¯
Ë
ˆ˜ ˜ˆ˜ ÁÊÁ ÊÁ 7
˜˜ ˜˜ ; 2 ÁÁ cos ÁÁ
˜˜ ˜˜ ÁÁ ÁÁ 4
¯¯ Ë Ë
ˆ˜
Ê
˜˜  i  sin ÁÁÁ 7
˜˜
ÁÁ 4
¯
Ë
PTS: 1
70. ANS:
ÁÊ  ˜ˆ ˆ˜˜
7 ÊÁÁ ÁÊ  ˜ˆ
14 ÊÁË cos ( )  i  sin  ˆ˜¯ ;  ÁÁÁ cos ÁÁÁÁ ˜˜˜˜  i  sin ÁÁÁÁ ˜˜˜˜ ˜˜˜ ;
2 Á Ë3¯
Ë 3 ¯ ˜¯
Ë
PTS: 1
4
ˆ˜ ˜ˆ˜
˜˜ ˜˜
˜˜ ˜˜
¯¯
ID: A
71. ANS:
Ê
128 ÁÁÁ 1 
Ë
ˆ
3  i ˜˜˜
¯
PTS: 1
72. ANS:
1
16
PTS: 1
5
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