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Checkpoint: Assess Your Understanding, pages 461–464
6.1
1. A 15-m ladder leans against a wall at an angle of 55° to the
horizontal ground. To the nearest tenth of a metre, how far from the
wall is the base of the ladder and how far up the wall does it reach?
15
m
Sketch a diagram.
y
P(x, y)
The length of the ladder is OP.
The x-coordinate of P is the distance of the
ladder from the base of the wall.
x ⴝ r cos U
Substitute: r ⴝ 15, U ⴝ 55°
55⬚
x
x ⴝ 15 cos 55°
O
x ⴝ 8.6036. . .
The base of the ladder is approximately 8.6 m
from the wall.
The y-coordinate of P is the distance that the ladder reaches up the wall.
y ⴝ r sin U
Substitute: r ⴝ 15, U ⴝ 55°
y ⴝ 15 sin 55°
y ⴝ 12.2872. . .
The ladder reaches approximately 12.3 m up the wall.
2. Multiple Choice Which statements are correct?
I. cos 60° + sin 30° = 1
III. cos 45° + sin 45° =
√
II. sin 60° + cos 30° =
2
√
3
cos 45°
IV.
= 1
sin 45°
A. only I and II
B. only III and IV
C. no statements
D. all statements
6.2
3. For each angle below:
i) Sketch it in standard position.
ii) Determine its reference angle.
iii) Identify the other angles from 0° to 360° that have the same
reference angle.
a) 205°
i) Since the angle is between 180° and 270°, the terminal arm lies
in Quadrant 3.
ii) Its reference angle is: 205° ⴚ 180° ⴝ 25°
iii) In Quadrant 2, the angle with the same reference angle is:
180° ⴚ 25° ⴝ 155°
In Quadrant 4, the angle with the same reference angle is:
360° ⴚ 25° ⴝ 335°
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Chapter 6: Trigonometry—Checkpoint—Solutions
y
205⬚
x
O
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b) 12°
i) Since the angle is between 0° and 90°, the terminal arm is in
Quadrant 1.
ii) Its reference angle is: 12°
iii) In Quadrant 2, the angle with the same reference angle is:
180° 12° 168°
In Quadrant 3, the angle with the same reference angle is:
180° 12° 192°
In Quadrant 4, the angle with the same reference angle is:
360° 12° 348°
y
12⬚ x
O
4. The point P(11, -7) is on the terminal arm of angle u in
standard position.
a) Sketch the angle.
y
Plot P(11, 7); draw a line through OP.
Label U.
x
b) Determine the primary trigonometric ratios of u.
Let the length of OP r.
√2
Substitute: x 11, y 7
x y2
√ 2
2
r (11) ( 7)
√
r 170
√
Use: r x 11, y 7, r y
sin U r
y
x
11
√
170
P(11, ⫺7)
170
cos U r
7
√
␪
O
170
tan U x
7
11
c) To the nearest degree, what is u?
11
Use: cos U √
170
The reference angle is:
cos1 a√
11
b 32.4711. . .°
170
In Quadrant 4, U is approximately: 360° 32° 328°
4
5. To the nearest degree, which angles satisfy the equation cos u = - 7
for 0° ≤ u ≤ 360°?
Use: cos U 4
7
The reference angle is:
cos1 a b ⬟ 55°
4
7
In Quadrant 2, U is approximately: 180° 55° 125°
In Quadrant 3, U is approximately: 180° 55° 235°
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6. a) Sketch a diagram to show these angles in standard position:
0°;
90°;
180°;
270°;
360°
b) Without using technology, determine the sine, cosine, and
tangent of each angle in part a.
For 0°, the terminal arm is on the positive x-axis, so x r and y 0.
y
y
180⬚ 90⬚
O
270⬚
0⬚ x
360⬚
y
Use: sin U r
cos U r
x
tan U x
0
r
0
tan 0° r
sin 0° r
cos 0° r
0
1
0
For 90°, the terminal arm is on the positive y-axis, so x 0 and y r.
0
r
r
tan 90° , which is undefined
sin 90° r
cos 90° r
0
1
0
For 180°, the terminal arm is on the negative x-axis, so x r and y 0.
r
0
0
tan 180° r
sin 180° r
cos 180° r
0
1
0
For 270°, the terminal arm is on the negative y-axis, so x 0 and
y r.
r
0
r
tan 270° sin 270° r
cos 270° r
, which is undefined
0
1
0
For 360°, the terminal arm is on the positive x-axis, so x r and y 0.
0
r
sin 360° r
0
cos 360° r
1
0
tan 360° r
0
7. Determine the exact primary trigonometric ratios for each angle in
standard position.
a) 135°
The terminal arm of the angle lies in Quadrant 2, and its reference
angle is: 180° 135° 45°
sin 135° sin 45°
cos 135° cos 45°
1
1
√
√
2
2
tan 135° tan 45°
1
b) 300°
The terminal arm of the angle lies in Quadrant 4, and its reference
angle is: 360° 300° 60°
sin 300° sin 60°
cos 300° cos 60°
√
3
2
1
2
tan 300° √
tan 60°
3
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c) 210°
The terminal arm of the angle lies in Quadrant 3, and its reference
angle is: 210° 180° 30°
sin 210° sin 30°
cos 210° cos
30°
√
1
2
3
2
tan 210° tan 30°
1
√
3
8. Angle u is in standard position and its terminal arm lies in
1
Quadrant 4. The cosine of its reference angle is 8 . Determine the
exact values of sin u, cos u, and tan u.
For the reference angle in Quadrant 1:
Use: r2 x2 y2
Substitute: x 1, r 8
(8)2 (1)2 y 2
y2 63
y
√
63
√
In Quadrant 4, x 1, y 63, r 8
y
Use: sin U r
x
√
63
8
cos U r
y
tan U x
√
√
63
, or 63
1
1
8
3
9. Multiple Choice Angle u is in standard position, with tan u = - 2 .
Which statement could be correct?
3
13
A. sin u = √
2
C. sin u = - √
13
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3
13
B. cos u = - √
3
13
D. cos u = √
Chapter 6: Trigonometry—Checkpoint—Solutions
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