Name: ________________________ Class: ___________________ Date: __________
ID: A
Unit two review (trig)
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1. What is the reference angle for 15° in standard position?
A 255°
C 345°
B 30°
D 15°
____
2. What is the exact sine of ∠A?
A
B
1/ 3
1/3
C
D
2/ 3
1/2
____
3. Which set of angles has the same terminal arm as 40°?
A 80°, 120°, 160°
C 200°, 380°, 560°
B 130°, 220°, 310°
D 400°, 760°, 1120°
____
4. The point (40, –9) is on the terminal arm of ∠A. Which is the set of exact primary trigonometric ratios for
the angle?
41
41
9
A sin A = − , cos A = , tan A = −
9
40
40
40
9
40
B sin A = , cos A = − , tan A = −
41
41
9
40
9
9
C sin A = − , cos A = , tan A = −
41
41
40
9
40
9
D sin A = − , cos A = , tan A = −
41
41
40
____
5. Marco is 450 m due east of the centre of the park. His friend Ray is 450 m due south of the centre of the park.
Which is the correct expression for the exact distance between the two boys?
A 225 2 m
C 450 2 m
225
450
B
m
D
m
2
2
1
Name: ________________________
ID: A
____
6. Solve to the nearest tenth of a unit for the unknown side in the ratio
a
12
=
.
sin30° sin115°
A 24
C 6.6
B 21.8
D 24.6
____
7. Determine the length of x, to the nearest tenth of a centimetre.
A
B
____
26.6
36.5
C
D
11.2
17.1
8. Determine, to the nearest tenth of a centimetre, the two possible lengths of a.
A
B
72.8 cm and 26.3 cm
34.3 cm and 26.3 cm
C
D
2
72.8 cm and 55.8 cm
55.8 cm and 34.3 cm
Name: ________________________
____
ID: A
9. Determine, to the nearest tenth of a degree, the two possible measures of ∠C.
A
B
9° and 81°
161.6° and 171°
C
D
3
161.6° and 0.4°
9° and 171°
Name: ________________________
ID: A
____ 10. Which of the following triangles cannot be solved using the sine law?
Diagrams not drawn to scale.
A
C
B
D
4
Name: ________________________
ID: A
____ 11. If ∠B = 58.8°, c = 10.3 cm, and b = 10.5 cm, and ∆ABC is acute, what is the measure of ∠C, to the nearest
tenth of a degree?
A
B
57°
123.0°
C
D
30.5°
149.5°
____ 12. Which strategy would be best to solve for x in the triangle shown?
A
B
cosine law
primary trigonometric ratios
C
D
5
sine law
none of the above
Name: ________________________
ID: A
____ 13. Determine the measure of x, to the nearest tenth of a degree.
A
B
25.6°
18.1°
C
D
136.3°
71.9°
____ 14. What is the length of x, to the nearest tenth of a metre?
A
B
27.7 m
21.8 m
C
D
6
26.1 m
37.6 m
Name: ________________________
ID: A
____ 15. If ∠Q = 31°, r = 20 cm, and p = 23 cm, what is the length of q, to the nearest centimetre?
A
B
21 cm
30 cm
C
D
12 cm
11 cm
____ 16. Solve the following triangle, rounding side lengths to the nearest tenth of a unit and angle measures to the
nearest degree.
Diagram not drawn to scale.
∠A = 152°, b = 19, a = 23.5
A
B
C
D
∠B = 22°, ∠C = 6°, c = 5.0
∠B = 158°, ∠C = 84°, c = 5.0
∠B = 68°, ∠C = 174°, c = 28.7
∠B = 35°, ∠C = 7°, c = 28.2
7
Name: ________________________
ID: A
____ 17. While flying, a helicopter pilot spots a water tower that is 7.4 km to the north. At the same time, he sees a
monument that is 8.5 km to the south. The tower and the monument are separated by a distance of 11.4 km
along the flat ground. What is the angle made by the water tower, helicopter, and monument?
A
B
91°
11°
C
D
40°
48°
Completion
Complete each statement.
1. The expression cos 30° is equivalent to sin ____________________.
2. An angle between 0° and 360° that has the same sine value as sin 133° is ____________________.
3. The tangent ratio is positive in the first and ____________________ quadrants.
4. The ____________________ law is used to solve a triangle when two sides and their contained angle are
given.
5. The sine law for acute
PQR states that ∠Q ____________________.
Short Answer
1. a) For the given trigonometric ratio, determine two other angles that give the same value.
i) sin 45°
ii) tan 300°
iii) cos 120°
b) Explain how you determined the angles in part a).
8
Name: ________________________
ID: A
Problem
12
.
13
a) In which quadrant(s) is this angle? Explain.
b) If the sine of the angle is negative, in which quadrant is the angle? Explain.
c) Sketch a diagram to represent the angle in standard position, given that the condition in part b) is true.
d) Find the coordinates of a point on the terminal arm of the angle.
e) Write exact expressions for the other two primary trigonometric ratios for the angle.
1. Consider ∠A such that cos A =
2. Two wires are connected to a tower at the same point on the tower. Wire 1 makes an angle of 45° with the
ground and wire 2 makes an angle of 60° with the ground.
a) Represent this situation with a diagram.
b) Which wire is longer? Explain.
c) If the point where the two wires connect to the tower is 35 m above the ground, determine exact
expressions for the lengths of the two wires.
d) Determine the length of each wire, to the nearest tenth of a metre.
e) How do your answers to parts b) and d) compare?
9
ID: A
Unit two review (trig)
Answer Section
MULTIPLE CHOICE
1. ANS:
NAT:
2. ANS:
NAT:
3. ANS:
NAT:
4. ANS:
NAT:
KEY:
5. ANS:
NAT:
KEY:
6. ANS:
NAT:
7. ANS:
NAT:
8. ANS:
NAT:
KEY:
9. ANS:
NAT:
KEY:
10. ANS:
NAT:
11. ANS:
NAT:
12. ANS:
NAT:
13. ANS:
NAT:
14. ANS:
NAT:
15. ANS:
NAT:
16. ANS:
NAT:
KEY:
17. ANS:
NAT:
D
PTS: 1
DIF: Easy
T1
TOP: Angles in Standard Position
D
PTS: 1
DIF: Average
T1
TOP: Angles in Standard Position
D
PTS: 1
DIF: Easy
T1
TOP: Angles in Standard Position
D
PTS: 1
DIF: Average
T1
TOP: Trigonometric Ratios of Any Angle
point on terminal arm | cosine | sine | tangent
D
PTS: 1
DIF: Easy
T1
TOP: Trigonometric Ratios of Any Angle
tangent
C
PTS: 1
DIF: Easy
T3
TOP: The Sine Law
B
PTS: 1
DIF: Easy
T3
TOP: The Sine Law
B
PTS: 1
DIF: Difficult
T3
TOP: The Sine Law
sine law | side length | ambiguous case
C
PTS: 1
DIF: Difficult
T3
TOP: The Sine Law
sine law | angle measure | ambiguous case
A
PTS: 1
DIF: Average
T3
TOP: The Sine Law
A
PTS: 1
DIF: Average
T3
TOP: The Sine Law
C
PTS: 1
DIF: Easy
T3
TOP: The Sine Law
B
PTS: 1
DIF: Average
T3
TOP: The Cosine Law
C
PTS: 1
DIF: Average
T3
TOP: The Cosine Law
C
PTS: 1
DIF: Average
T3
TOP: The Cosine Law
A
PTS: 1
DIF: Difficult
T3
TOP: The Sine Law | The Cosine Law
cosine law | sine law | solve a triangle
A
PTS: 1
DIF: Average
T3
TOP: The Cosine Law
1
OBJ:
KEY:
OBJ:
KEY:
OBJ:
KEY:
OBJ:
Section 2.1
reference angle | < 180°
Section 2.1
special angles | sine
Section 2.1
co-terminal angles
Section 2.2
OBJ: Section 2.2
OBJ:
KEY:
OBJ:
KEY:
OBJ:
Section 2.3
sine law | side length
Section 2.3
sine law | side length
Section 2.3
OBJ: Section 2.3
OBJ:
KEY:
OBJ:
KEY:
OBJ:
KEY:
OBJ:
KEY:
OBJ:
KEY:
OBJ:
KEY:
OBJ:
Section 2.3
sine law | < 180°
Section 2.3
sine law | angle measure
Section 2.3
sine law | solution method
Section 2.4
cosine law | angle measure
Section 2.4
cosine law | side length
Section 2.4
cosine law | side length
Section 2.3 | Section 2.4
OBJ: Section 2.4
KEY: cosine law | angle measure
ID: A
COMPLETION
1. ANS: 60°
PTS: 1
DIF: Easy
TOP: Angles in Standard Position
2. ANS: 47°
OBJ: Section 2.1 NAT: T 1
KEY: cosine | sine | special angles
PTS: 1
DIF: Average
OBJ: Section 2.2
TOP: Trigonometric Ratios of Any Angle
3. ANS: third or 3rd
NAT: T 1
KEY: sine | reference angle
PTS: 1
DIF: Easy
OBJ: Section 2.2
TOP: Trigonometric Ratios of Any Angle
4. ANS: cosine
NAT: T 2
KEY: ratio | quadrant
PTS: 1
NAT: T 3
DIF: Average
OBJ: Section 2.3 | Section 2.4
TOP: The Sine Law | The Cosine Law
KEY: cosine law | sine law
Ê q sinR ˜ˆ
Ê q sinP ˜ˆ
Á
Á
Á
˜˜
Á
˜˜
˜ or sin −1 ÁÁÁ
˜
5. ANS: sin −1 ÁÁÁ
Á r ˜˜
Á p ˜˜
Ë
¯
Ë
¯
PTS: 1
DIF:
TOP: The Sine Law
Average
OBJ: Section 2.3
KEY: sine law
NAT: T 3
SHORT ANSWER
1. ANS:
a) Answers may vary. Sample answers:
i) –315° and 405°
ii) –60° and 120°
iii) 240° and –240°
b) Sketch the given angle on a Cartesian plane, and identify its reference angle. Then, determine the other
quadrant where the trigonometric ratio has the same sign as the given ratio and reflect the reference angle
into that quadrant. Any angle co-terminal to the two angles in the diagram will have the same trigonometric
ratio as that given.
PTS: 1
DIF: Easy
OBJ: Section 2.1 | Section 2.2
NAT: T 1 | T 2
TOP: Angles in Standard Position | Trigonometric Ratios of Any Angle
KEY: primary trigonometric ratios | reference angle
2
ID: A
PROBLEM
1. ANS:
a) Since the cosine ratio is positive, the angle is in the first or the fourth quadrant.
b) If the sine ratio is negative, the angle is located in the fourth quadrant.
c)
d) Use the Pythagorean theorem.
r2 = x2 + y2
13 2 = 12 2 + y 2
y 2 = 169 − 144
= 25
y = ±5
Since the point is in the fourth quadrant, y = –5.
Therefore, a point on the terminal arm is (12, –5).
5
5
e) sinA = − ,tanA = −
13
12
PTS: 1
DIF: Average
OBJ: Section 2.1 | Section 2.2
NAT: T 1 | T 2
TOP: Angles in Standard Position | Trigonometric Ratios of Any Angle
KEY: primary trigonometric ratios | standard angle
3
ID: A
2. ANS:
a)
b) Both wires are connected to the tower at the same height, which is the opposite side to the given angles.
Each wire length represents the hypotenuse of its respective triangle. The longer hypotenuse is the wire that
forms the smaller angle, as it will need to be longer to reach the tower.
c) Let x represent the length of wire 1 and y represent the length of wire 2.
35
35
Wire 2: sin60° =
Wire 1: sin45° =
x
y
x=
35
sin45°
35
= Ê
ÁÁ 1 ˆ˜˜
ÁÁ
˜˜
ÁÁ
˜˜
2
Ë
¯
= 35 2
y=
35
sin60°
35
= Ê
ÁÁ 3 ˆ˜˜
ÁÁ
˜˜
ÁÁ
˜
ÁÁ 2 ˜˜˜
Ë
¯
70
3
d) The length of wire 1 is 49.5 m, and the length of wire 2 is 40.4 m.
e) The values calculated in part d) support the answers in part b).
=
PTS: 1
DIF: Average
OBJ: Section 2.1 | Section 2.2
NAT: T 1 | T 2
TOP: Angles in Standard Position | Trigonometric Ratios of Any Angle
KEY: special angles | sine
4
Unit two review (trig) [Answer Strip]
C
_____
6.
D
_____
1.
C
_____
9.
ID: A
A 10.
_____
A 11.
_____
B
_____
7.
D
_____
2.
C 12.
_____
B
_____
8.
D
_____
3.
D
_____
4.
D
_____
5.
Unit two review (trig) [Answer Strip]
B 13.
_____
C 15.
_____
C 14.
_____
A 16.
_____
A 17.
_____
ID: A