Download ExamView - A2TU12 - Review Problem Set.tst

Name: ________________________ Class: ___________________ Date: __________
ID: A
A2T Unit 12 - Intro to Trigonometry Review
Multiple Choice
Identify the choice that best completes the statement or answers the question.
1. An angle, P, drawn in standard position, terminates
in Quadrant II if
a. cos P < 0 and csc P < 0
b. sinP > 0 and cos P > 0
c. csc P > 0 and cot P < 0
d. tanP < 0 and sec P > 0
5. The expression
a.
b.
c.
d.
2. If sin θ > 0 and sec θ < 0, in which quadrant does
the terminal side of angle θ lie?
a. I
b. II
c. III
d. IV
sin θ
cos θ
sin θ
cos θ
cos θ
sin θ
6. The expression
a.
b.
c.
d.
3. If the tangent of an angle is negative and its secant
is positive, in which quadrant does the angle
terminate?
a. I
b. II
c. III
d. IV
sec θ
is equivalent to
csc θ
tan θ
is equivalent to
sec θ
cos 2 θ
sin θ
sin θ
cos 2 θ
cos θ
sin θ
7. The expression cot θ ⋅ sec θ is equivalent to
cos θ
a.
sin 2 θ
sin θ
b.
cos 2 θ
c. csc θ
d. sin θ
4. If sin θ is negative and cos θ is negative, in which
quadrant does the terminal side of θ lie?
a. I
b. II
c. III
d. IV
1
Name: ________________________
ID: A
10. For all values of θ for which the expression is
csc θ
defined,
is equivalent to
sec θ
a. cos θ
b. sin θ
c. cot θ
d. tan θ
8. The expression (sec 2 θ)(cot 2 θ)(sin θ) is equivalent
to
a. sin θ
b. cos θ
c. csc θ
d. sec θ
9. Expressed in simplest form, csc θ ⋅ tan θ ⋅ cos θ is
equivalent to
a. 1
b. sin θ
c. cos θ
d. tan θ
Short Answer
11. If the terminal side of angle θ passes through point
(−4,3), what is the value of cos θ ?
15. What is the number of degrees in an angle whose
8π
radian measure is
?
5
12. What is the number of degrees in an angle whose
11π
radian measure is
?
12
16. What is the number of degrees in an angle whose
7π
radian measure is
?
12
13. What is the radian measure of an angle whose
measure is −420°?
14. What is the number of degrees in an angle whose
measure is 2 radians?
2
Name: ________________________
ID: A
20. The pendulum of a clock swings through an angle
of 2.5 radians as its tip travels through an arc of 50
centimeters. Find the length of the pendulum, in
centimeters.
17. Cities H and K are located on the same line of
longitude and the difference in the latitude of these
cities is 9°, as shown in the accompanying diagram.
If Earth’s radius is 3,954 miles, how many miles
north of city K is city H along arc HK? Round your
answer to the nearest tenth of a mile.
21. As shown in the accompanying diagram, a dial in
the shape of a semicircle has a radius of 4
centimeters. Find the measure of θ, in radians,
when the pointer rotates to form an arc whose
length is 1.38 centimeters.
18. An arc of a circle that is 6 centimeters in length
intercepts a central angle of 1.5 radians. Find the
number of centimeters in the radius of the circle.
19. Kathy and Tami are at point A on a circular track
that has a radius of 150 feet, as shown in the
accompanying diagram. They run
counterclockwise along the track from A to S, a
distance of 247 feet. Find, to the nearest degree,
the measure of minor arc AS.
22. On the unit circle shown in the diagram below,
sketch an angle, in standard position, whose degree
measure is 240 and find the exact value of sin240°.
3
Name: ________________________
ID: A
23. If θ is an angle in standard position and its terminal
side passes through the point (−3,2), find the exact
value of csc θ .
30. Express the product of cos 30° and sin 45° in
simplest radical form.
ˆ
ÁÊÁ 1
3 ˜˜˜˜
Á
24. Point A ÁÁÁ ,
˜ is on the unit circle whose center
ÁÁ 2 2 ˜˜˜
Ë
¯
is the origin. If θ is an angle in standard position
whose terminal ray passes through point A, what is
the value of sin θ ?
31. Find the value of cos
5π
.
3
x
32. If f(x) = sinx + cos , find f(2π ).
2
25. Express sin (−170°) as a function of a positive acute
angle.
33. Find the value of sin 2
26. Express cos (−155°) as a function of a positive
acute angle.
34. Find the value of sin
27. Express sin (−215°) as a function of a positive acute
angle.
35. If f (x ) = 3cos x, find the numerical value of f (π ) .
π
3
π
2
.
− cos
36. Find the value of tan120° .
28. Express tan230° as a function of a positive acute
angle.
37. Find the value of tan(−135°) .
29. Express tan (−140°) as a function of a positive
acute angle.
ÊÁ π
38. If f(x) = 2 cos x, find f ÁÁÁ
ÁË 3
4
ˆ˜
˜˜ .
˜˜
¯
3π
.
2
Name: ________________________
ID: A
39. What is the numerical value of the product
ÊÁ
ˆÊ
ˆ
ÁÁ tan π ˜˜˜ ÁÁÁ cos π ˜˜˜ ?
ÁÁ
˜
Á
4 ˜¯ ÁË
3 ˜˜¯
Ë
ÊÁ π
40. If f (x ) = cos 2x, find f ÁÁÁ
ÁË 2
47. Find the exact value of csc
42. Find the exact value of sec
43. Find the exact value of cot
44. Find the exact value of cot
45. Find the exact value of csc
46. Find the exact value of sec
2
5
and sin θ < 0, find the remaining
13
five trigonometric values
48. If the cos θ = −
ˆ˜
˜˜ .
˜˜
¯
41. Find the exact value of csc
π
49. If the csc θ = −2 and cos θ > 0, find the remaining
five trigonometric values
π
6
π
6
π
6
π
4
π
4
π
2
5
ID: A
A2T Unit 12 - Intro to Trigonometry Review
Answer Section
MULTIPLE CHOICE
1. ANS: C
If csc P > 0, sin P > 0. If cot P < 0 and sin P > 0, cos P < 0
PTS: 2
REF: 061320a2
STA: A2.A.60
TOP: Finding the Terminal Side of an Angle
2. ANS: B
If sin θ > 0, then the terminal side of θ lies in either Quadrant I or II. If sec θ < 0, then cos θ < 0 and the
terminal side of θ lies in either Quadrant II or III.
PTS: 2
3. ANS: D
REF: 060302b
STA: A2.A.60
TOP: Finding the Terminal Side of an Angle
If the secant of an angle is positive, the cosine of the angle is positive and the terminal side of the angle
lies in either Quadrant I or IV. If the tangent of an angle is negative, then the signs of the cosine and sine
of that angle must be opposite. Since the cosine of the angle is positive, the sine of the angle must be
negative and the terminal side of the angle lies in either Quadrant III or IV.
PTS: 2
4. ANS: C
REF: 080410b
STA: A2.A.60
TOP: Finding the Terminal Side of an Angle
If sin θ is negative, the terminal side of θ lies in either Quadrant III or IV. If cos θ is negative, the
terminal side of θ lies in either Quadrant II or III.
PTS: 2
5. ANS: C
REF: 060502b
STA: A2.A.60
TOP: Finding the Terminal Side of an Angle
PTS: 2
6. ANS: D
REF: 010402b
STA: A2.A.58
TOP: Reciprocal Trigonometric Relationships
PTS: 2
7. ANS: C
REF: 010508b
STA: A2.A.58
TOP: Reciprocal Trigonometric Relationships
PTS: 2
REF: 010915b
STA: A2.A.58
TOP: Reciprocal Trigonometric Relationships
1
ID: A
8. ANS:
TOP:
9. ANS:
TOP:
10. ANS:
TOP:
C
PTS: 2
REF: 010427siii
Reciprocal Trigonometric Relationships
A
PTS: 2
REF: 069921siii
Reciprocal Trigonometric Relationships
C
PTS: 2
REF: 080318siii
Reciprocal Trigonometric Relationships
STA: A2.A.58
STA: A2.A.58
STA: A2.A.58
SHORT ANSWER
11. ANS:
4
−
5
x
cos θ =
x2 + y2
=
−4
=−
(−4) 2 + 3 2
4
5
PTS: 2
12. ANS:
165
11π 180
⋅
= 165
12
π
REF: 068628siii
STA: A2.A.62
TOP: Determining Trigonometric Functions
PTS: 2
KEY: degrees
13. ANS:
7π
−
3
ÊÁ π ˆ˜
˜˜ = − 7π
−420 ÁÁÁÁ
˜˜
180
3
Ë
¯
REF: 061002a2
STA: A2.M.2
TOP: Radian Measure
PTS: 2
KEY: radians
14. ANS:
360
REF: 081002a2
STA: A2.M.2
TOP: Radian Measure
REF: 011220a2
STA: A2.M.2
TOP: Radian Measure
REF: 061302a2
STA: A2.M.2
TOP: Radian Measure
π
2⋅
180
π
=
360
π
PTS: 2
KEY: degrees
15. ANS:
288
8π 180
⋅
= 288
5
π
PTS: 2
KEY: degrees
2
ID: A
16. ANS:
105.
PTS: 2
KEY: degrees
17. ANS:
REF: 080623b
. s=θr=
621.1.
PTS: 2
KEY: arc length
18. ANS:
STA: A2.M.2
π
20
TOP: Radian Measure
⋅ 3954 ≈ 621.1.
REF: 080426b
STA: A2.A.61
TOP: Arc Length
REF: 010526b
STA: A2.A.61
TOP: Arc Length
REF: 060531b
STA: A2.A.61
TOP: Arc Length
REF: 060626b
STA: A2.A.61
TOP: Arc Length
REF: 010725b
STA: A2.A.61
TOP: Arc Length
4.
PTS: 2
KEY: radius
19. ANS:
94.
PTS: 4
KEY: angle
20. ANS:
.
.
20.
PTS: 2
KEY: radius
21. ANS:
0.345.
PTS: 2
KEY: angle
3
ID: A
22. ANS:
−
PTS: 2
23. ANS:
13
. sin θ =
2
3
2
REF: 061033a2
y
=
x2 + y2
PTS: 2
24. ANS:
2
(−3) 2 + 2 2
STA: A2.A.60
=
2
. csc θ =
13
TOP: Unit Circle
13
.
2
REF: fall0933a2
STA: A2.A.62
TOP: Determining Trigonometric Functions
PTS: 2
25. ANS:
−sin 10° or −cos 80°
REF: 018514siii
STA: A2.A.62
TOP: Determining Trigonometric Functions
PTS: 2
26. ANS:
−sin 65° or −cos 25°
REF: 068014siii
STA: A2.A.57
TOP: Reference Angles
PTS: 2
27. ANS:
sin 35° or cos 55°
REF: 088416siii
STA: A2.A.57
TOP: Reference Angles
PTS: 2
28. ANS:
tan50° or cot 40°
REF: 068617siii
STA: A2.A.57
TOP: Reference Angles
PTS: 2
29. ANS:
tan40° or cot 50°
REF: 068811siii
STA: A2.A.57
TOP: Reference Angles
REF: 068912siii
STA: A2.A.57
TOP: Reference Angles
3
2
PTS: 2
4
ID: A
30. ANS:
2
6
3
×
=
2
2
4
PTS: 2
REF: 061331a2
KEY: degrees, common angles
31. ANS:
0.5
STA: A2.A.56
TOP: Determining Trigonometric Functions
PTS: 2
REF: 068018siii
KEY: radians, other angles
32. ANS:
−1
STA: A2.A.56
TOP: Determining Trigonometric Functions
PTS: 2
REF: 010408siii
KEY: radians, common angles
33. ANS:
3
4
STA: A2.A.56
TOP: Determining Trigonometric Functions
PTS: 2
REF: 018402siii
KEY: radians, common angles
34. ANS:
1
STA: A2.A.56
TOP: Determining Trigonometric Functions
PTS: 2
REF: 068415siii
KEY: radians, common angles
35. ANS:
−3
STA: A2.A.56
TOP: Determining Trigonometric Functions
PTS: 2
REF: 018511siii
KEY: radians, common angles
36. ANS:
− 3
STA: A2.A.56
TOP: Determining Trigonometric Functions
PTS: 2
REF: 018517siii
KEY: degrees, other angles
37. ANS:
1
STA: A2.A.56
TOP: Determining Trigonometric Functions
PTS: 2
REF: 068709siii
KEY: degrees, other angles
38. ANS:
1
STA: A2.A.56
TOP: Determining Trigonometric Functions
PTS: 2
REF: 088711siii
KEY: radians, common angles
STA: A2.A.56
TOP: Determining Trigonometric Functions
5
ID: A
39. ANS:
1
2
PTS: 2
REF: 068814siii
KEY: radians, common angles
40. ANS:
−1
STA: A2.A.56
TOP: Determining Trigonometric Functions
PTS: 2
REF: 019012siii
KEY: radians, common angles
41. ANS:
2
STA: A2.A.56
TOP: Determining Trigonometric Functions
PTS: 1
42. ANS:
STA: A2.A.59
2 3
3
PTS: 1
43. ANS:
3
STA: A2.A.59
PTS: 1
44. ANS:
1
STA: A2.A.59
PTS: 1
45. ANS:
2
STA: A2.A.59
PTS: 1
46. ANS:
undefined
STA: A2.A.59
PTS: 1
47. ANS:
0
STA: A2.A.59
PTS: 1
STA: A2.A.59
6
ID: A
48. ANS:
sin θ =
12
13
tan θ = −
12
5
csc θ = −
13
5
sec θ = −
13
5
cot θ = −
12
5
PTS: 1
49. ANS:
sin θ = −
STA: A2.A.58
1
2
tan θ = −
sec θ = −
3
3
13
5
cot θ = − 3
cos θ =
PTS: 1
3
2
STA: A2.A.58
7