08_ch06_pre-calculas12_wncp_solution.qxd
5/21/12
8:33 PM
Page 18
Home
Quit
Checkpoint: Assess Your Understanding, pages 505–509
6.1
1. a) Sketch each angle u in standard position.
i) u ⫽ ⫺405°
ii) u ⫽ 760°
y
O
y
x
405°
x
O
760°
b) For each angle u in part a
• Determine the measures of angles that are coterminal with u
in the domain ⫺800° ≤ u ≤ 800°.
• Write an expression for all angles that are coterminal with u.
i) The measures of the angles
coterminal with ⴚ405°
are: ⴚ405°;
ⴚ405° ⴙ 360° ⴝ ⴚ45°;
ⴚ405° ⴙ 2(360°) ⴝ 315°;
ⴚ405° ⴙ 3(360°) ⴝ 675°;
and ⴚ405° ⴚ 360° ⴝ ⴚ765°
The measures of all angles
coterminal with ⴚ405° are:
ⴚ405° ⴙ k360°, k ç ⺪
ii) The measures of the angles
coterminal with 760° are: 760°;
760° ⴚ 360° ⴝ 400°;
760° ⴚ 2(360°) ⴝ 40°;
760° ⴚ 3(360°) ⴝ ⴚ320°;
and 760° ⴚ 4(360°) ⴝ ⴚ680°
The measures of all angles
coterminal with 760° are:
760° ⴙ k360°, k ç ⺪
2. Determine the exact value of each trigonometric ratio.
a) cos 30°
√
ⴝ
3
2
d) sin (⫺300°)
ⴝ sin 60º
√
ⴝ
18
3
2
b) tan (⫺60°)
c) csc 90°
ⴝ ⴚtan 60°
√
ⴝⴚ 3
e) sec 225°
ⴝ
1
sin 90°
ⴝ1
f) cot 495°
1
cos 225°
1
ⴝⴚ
cos 45°
1
tan 495°
1
ⴝ
tan 135°
ⴝ
ⴝ
√
ⴝⴚ 2
ⴝ ⴚ1
Chapter 6: Trigonometry—Checkpoint—Solutions
DO NOT COPY. ©P
08_ch06_pre-calculas12_wncp_solution.qxd
5/21/12
8:33 PM
Page 19
Home
Quit
3. P(2, ⫺3) is a terminal point of angle u in standard position. To the
nearest degree, determine possible values of u in the domain
⫺360° ≤ u ≤ 360°.
The terminal arm of angle U is in Quadrant 4.
3
2
tan U ⴝ ⴚ , or ⴚ1.5
The reference angle is: tanⴚ1(1.5) ⴝ 56.3099. . .º
So, U ⴝ 360° ⴚ 56.3099. . .º
ⴝ 303.6900. . .º
Another value of U in the given domain is: ⴚ56.3099. . .º
To the nearest degree, U is ⴚ56º or 304º.
1
4. Given cos u ⫽ 3
a) Determine the exact values of the other trigonometric ratios for
180° ≤ u ≤ 360°.
Let P(x, y) on a circle, radius r, be the terminal point of angle U in
standard position.
x
Then, cos U ⴝ r
1
x
r ⴝ , so choose x ⴝ 1 and r ⴝ 3
3
Use: x2 ⴙ y2 ⴝ r2
Substitute for x and r.
1 ⴙ y2 ⴝ 9
√
yⴝ— 8
Since cos U is positive, for the given domain the terminal arm of U lies
√
in Quadrant 4 where y is negative; so y ⴝ ⴚ 8
√
√
8
3
1
sec U ⴝ 3 sin U ⴝ ⴚ
csc U ⴝ ⴚ √ tan U ⴝ ⴚ 8 cot U ⴝ ⴚ √
3
8
8
b) To the nearest degree, determine all possible values of u in the
domain ⫺360° ≤ u ≤ 360°.
cos U ⴝ , so the reference angle is: cosⴚ1 a b ⴝ 70.5287. . .º
1
3
1
3
So, U ⴝ ⴚ70.5287. . .º or U ⴝ 360º ⴚ 70.5287. . .º
ⴝ 289.4712. . .º
To the nearest degree, U is ⴚ71º or 289º.
5. Multiple Choice Point P(⫺2, ⫺1) lies on the terminal arm of
angle u in standard position. Which statement is correct?
1
A. tan u ⫽ 2
1
C. cos u ⫽ - √
5
©P DO NOT COPY.
2
5
B. sin u ⫽ - √
√
5
D. csc u ⫽ 2
Chapter 6: Trigonometry—Checkpoint—Solutions
19
08_ch06_pre-calculas12_wncp_solution.qxd
5/21/12
8:33 PM
Page 20
Home
Quit
6.2
6. In terms of ␲, determine the length of the arc that subtends a
central angle of 240° in a circle with radius 9 cm.
Arc length is:
240
(2␲)(9) ⴝ 12␲
360
The arc length is 12␲ cm.
6.3
7. a) Sketch each angle u in standard position.
13␲
i) u ⫽ ⫺3␲
ii) u ⫽ 6
y
y
x
x
O
0
3
13
6
b) For each angle u in part a
• Determine the measures of angles that are coterminal with u
in the domain ⫺4␲ ≤ u ≤ 4␲.
• Write an expression for all angles that are coterminal with u.
i) The measures of the angles
coterminal with ⴚ3␲ are:
ⴚ3␲; ⴚ3␲ ⴙ 2␲ ⴝ ⴚ␲;
ⴚ3␲ ⴙ 4␲ ⴝ ␲;
and ⴚ3␲ ⴙ 6␲ ⴝ 3␲
The measures of all angles
are: ⴚ3␲ ⴙ 2␲k, k ç ⺪
ii) The measures of the angles
coterminal with
13␲
ⴚ 2␲ ⴝ
6
13␲
ⴚ 4␲ ⴝ
6
13␲
ⴚ 6␲ ⴝ
6
13␲
13␲
are:
;
6
6
␲
;
6
11␲
; and
6
23␲
ⴚ
;
6
ⴚ
The measures of all angles are:
13␲
ⴙ 2␲k, k ç ⺪
6
8. Multiple Choice For which pair of values of u is sin u ⫽ ⫺cos u?
␲
3␲
3␲
5␲
A. u ⫽ 4 and u ⫽ 4
B. u ⫽ 4 and u ⫽ 4
5␲
7␲
3␲
7␲
C. u ⫽ 4 and u ⫽ 4
D. u ⫽ 4 and u ⫽ 4
20
Chapter 6: Trigonometry—Checkpoint—Solutions
DO NOT COPY. ©P
08_ch06_pre-calculas12_wncp_solution.qxd
5/21/12
8:33 PM
Page 21
Home
Quit
9. Determine the exact value of each trigonometric ratio.
␲
␲
11␲
a) tan 4
b) csc a- 6 b
c) sin 3
ⴝ1
ⴝⴚ
1
ⴝ ⴚsin
␲
sin
6
√
ⴝⴚ
ⴝ ⴚ2
8␲
13␲
d) sec a- 3 b
ⴝ sec aⴚ
ⴝ
11␲
ⴝ cos aⴚ
␲
6
ⴝ cos
1
tan
ⴝ
ⴝ ⴚ2
␲
6
ⴝ cot
ⴝ
1
2␲
cos
3
√
3
2
f) cos a- 4 b
e) cot 6
2␲
b
3
␲
3
3␲
b
4
3␲
4
1
ⴝ ⴚ√
2
3
10. P(⫺5, ⫺2) is a terminal point of angle u in standard position.
a) State the exact values of all trigonometric ratios for u.
Let the distance between the origin and P be r.
Substitute: x ⴝ ⴚ5, y ⴝ ⴚ2
Use: x2 ⴙ y2 ⴝ r2
25 ⴙ 4 ⴝ r2
√
r ⴝ 29
2
29
sin U ⴝ ⴚ √
5
29
cos U ⴝ ⴚ √
tan U ⴝ
2
5
√
29
2
√
29
sec U ⴝ ⴚ
5
csc U ⴝ ⴚ
cot U ⴝ
5
2
b) To the nearest tenth of a radian, determine all possible values of u
in the domain ⫺2␲ ≤ u ≤ 2␲.
The terminal arm of angle U lies in Quadrant 3.
The reference angle is: tanⴚ1 a b ⴝ 0.3805. . .
2
5
So, U ⴝ ␲ ⴙ 0.3805. . .
ⴝ 3.5220. . .
The angle between ⴚ2␲ and 0 that is coterminal with 3.5220. . . is:
ⴚ2␲ ⴙ 3.5220. . . ⴝ ⴚ2.7610. . .
The possible values of U are approximately: 3.5 and ⴚ2.7
©P DO NOT COPY.
Chapter 6: Trigonometry—Checkpoint—Solutions
21
08_ch06_pre-calculas12_wncp_solution.qxd
5/21/12
8:33 PM
Page 22
Home
Quit
11. a) Convert each angle to radians.
ii) ⫺115°
i) 450°
ⴝ 115 aⴚ
5␲
␲
b , or
180
2
ⴝ 450 a
␲
23␲
b , or ⴚ
180
36
b) Convert each angle to degrees. Give the exact answer where
possible. Write the answer to the nearest degree where necessary.
7␲
4␲
ii) ⫺ 7
i) 4
7
4
4
7
ⴝ (180°), or 315°
ⴝ (ⴚ180º), or approximately ⴚ103°
12. An arc of length 4.5 cm is marked on the circumference of a circle
with radius 10.0 cm. What is the area of the sector of the circle
formed by the arc and the radii that join the centre of the circle to
the endpoints of the arc?
Measure of central angle ⴝ
arc length
radius
4.5
ⴝ , or 0.45
10
The area of the sector is proportional to the central angle.
0.45
Area of sector
ⴝ
2␲
Area of circle
0.45␲(10)2
Area of sector ⴝ
cm2
2␲
ⴝ 22.5 cm2
22
Chapter 6: Trigonometry—Checkpoint—Solutions
DO NOT COPY. ©P