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Checkpoint: Assess Your Understanding, pages 505–509
6.1
1. a) Sketch each angle u in standard position.
i) u ϭ Ϫ405° ii) u ϭ 760°
y y
x x
O Ϫ405° O
760°
b) For each angle u in part a
• Determine the measures of angles that are coterminal with u
in the domain Ϫ800° ≤ u ≤ 800°.
• Write an expression for all angles that are coterminal with u.
i) The measures of the angles ii) The measures of the angles
coterminal with ؊405° coterminal with 760° are: 760°;
are: ؊405°; 760° ؊ 360° ‫;°004 ؍‬
؊405° ؉ 360° ‫;°54؊ ؍‬ 760° ؊ 2(360°) ‫;°04 ؍‬
؊405° ؉ 2(360°) ‫;°513 ؍‬ 760° ؊ 3(360°) ‫;°023؊ ؍‬
؊405° ؉ 3(360°) ‫;°576 ؍‬ and 760° ؊ 4(360°) ‫°086؊ ؍‬
and ؊405° ؊ 360° ‫°567؊ ؍‬ The measures of all angles
The measures of all angles coterminal with 760° are:
coterminal with ؊405° are: 760° ؉ k360°, k ç ‫ޚ‬
؊405° ؉ k360°, k ç ‫ޚ‬
2. Determine the exact value of each trigonometric ratio.
a) cos 30° b) tan (Ϫ60°) c) csc 90°
√
3 ‫؊ ؍‬tan 60° ‫؍‬
1
‫؍‬ √ sin 90°
2 ‫3 ؊؍‬
‫1؍‬
d) sin (Ϫ300°) e) sec 225° f) cot 495°
‫ ؍‬sin 60º 1 1
√
‫؍‬ ‫؍‬
cos 225° tan 495°
3 1
‫؍‬ ‫؊؍‬ ‫؍‬
1
2 cos 45° tan 135°
√
‫2 ؊؍‬ ‫1؊ ؍‬
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3. P(2, Ϫ3) is a terminal point of angle u in standard position. To the
nearest degree, determine possible values of u in the domain
Ϫ360° ≤ u ≤ 360°.
The terminal arm of angle U is in Quadrant 4.
3
tan U ‫ , ؊ ؍‬or ؊1.5
2
The reference angle is: tan؊1(1.5) ‫. . .9903.65 ؍‬º
So, U ‫. . .9903.65 ؊ °063 ؍‬º
‫. . .0096.303 ؍‬º
Another value of U in the given domain is: ؊56.3099. . .º
To the nearest degree, U is ؊56º or 304º.
1
4. Given cos u ϭ 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 ‫ ؍‬r
x 1
r ‫ , ؍‬so choose x ‫ 1 ؍‬and r ‫3 ؍‬
3
Use: x2 ؉ y2 ‫ ؍‬r2 Substitute for x and r.
1 ؉ y2 ‫9 ؍‬
√
y‫8 —؍‬
Since cos U is positive, for the given domain the terminal arm of U lies
√
in Quadrant 4 where y is negative; so y ‫8 ؊ ؍‬
√
8 3 √ 1
sec U ‫ 3 ؍‬sin U ‫؊ ؍‬ csc U ‫ √ ؊ ؍‬tan U ‫ 8 ؊ ؍‬cot U ‫√ ؊ ؍‬
3 8 8
b) To the nearest degree, determine all possible values of u in the
domain Ϫ360° ≤ u ≤ 360°.
cos U ‫ , ؍‬so the reference angle is: cos؊1 a b ‫. . .7825.07 ؍‬º
1 1
3 3
So, U ‫. . .7825.07؊ ؍‬º or U ‫063 ؍‬º ؊ 70.5287. . .º
‫. . .2174.982 ؍‬º
To the nearest degree, U is ؊71º or 289º.
5. Multiple Choice Point P(Ϫ2, Ϫ1) lies on the terminal arm of
angle u in standard position. Which statement is correct?
1 2
A. tan u ϭ 2 B. sin u ϭ - √
5
√
1 5
C. cos u ϭ - √ D. csc u ϭ 2
5
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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.
240
Arc length is: (2␲)(9) ‫␲21 ؍‬
360
The arc length is 12␲ cm.
6.3
7. a) Sketch each angle u in standard position.
13␲
i) u ϭ Ϫ3␲ ii) u ϭ 6
y y
x x
O 0 13␲
Ϫ3␲ 6
b) For each angle u in part a
• Determine the measures of angles that are coterminal with u
in the domain Ϫ4␲ ≤ u ≤ 4␲.
• Write an expression for all angles that are coterminal with u.
i) The measures of the angles ii) The measures of the angles
coterminal with ؊3␲ are: coterminal with
13␲
are:
13␲
;
؊3␲; ؊3␲ ؉ 2␲ ‫;␲؊ ؍‬ 6 6
13␲ ␲
؊3␲ ؉ 4␲ ‫;␲ ؍‬ ؊ 2␲ ‫؍‬ ;
6 6
and ؊3␲ ؉ 6␲ ‫␲3 ؍‬ 13␲ 11␲
؊ 4␲ ‫؍‬ ؊ ; and
The measures of all angles 6 6
are: ؊3␲ ؉ 2␲k, k ç ‫ޚ‬ 13␲
؊ 6␲ ‫؍‬ ؊
23␲
;
6 6
The measures of all angles are:
؉ 2␲k, k ç ‫ޚ‬
13␲
6
8. Multiple Choice For which pair of values of u is sin u ϭ Ϫcos u?
␲ 3␲ 3␲ 5␲
A. u ϭ 4 and u ϭ 4 B. u ϭ 4 and u ϭ 4
5␲ 7␲ 3␲ 7␲
C. u ϭ 4 and u ϭ 4 D. u ϭ 4 and u ϭ 4
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9. Determine the exact value of each trigonometric ratio.
␲ ␲ 11␲
a) tan 4 b) csc a- 6 b c) sin 3
‫1؍‬ 1 ␲
‫؊؍‬ ‫؊ ؍‬sin
␲ 3
sin √
6 3
‫؊؍‬
‫2؊ ؍‬ 2
8␲ 13␲ 11␲
d) sec a- 3 b e) cot 6 f) cos a- 4 b
␲
‫ ؍‬sec a؊ b
2␲ ‫ ؍‬cot ‫ ؍‬cos a؊ b
3␲
3 6 4
1
‫؍‬
1 ‫؍‬ ‫ ؍‬cos
3␲
2␲ ␲ 4
cos tan
3 6 1
√ ‫√؊ ؍‬
‫2؊ ؍‬ ‫؍‬ 3 2
10. P(Ϫ5, Ϫ2) 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.
Use: x2 ؉ y2 ‫ ؍‬r2 Substitute: x ‫ ,5؊ ؍‬y ‫2؊ ؍‬
25 ؉ 4 ‫ ؍‬r2
√
r ‫92 ؍‬
√
2 29
sin U ‫√ ؊ ؍‬ csc U ‫؊ ؍‬
29 2
√
5 29
cos U ‫√ ؊ ؍‬ sec U ‫؊ ؍‬
29 5
2 5
tan U ‫؍‬ cot U ‫؍‬
5 2
b) To the nearest tenth of a radian, determine all possible values of u
in the domain Ϫ2␲ ≤ u ≤ 2␲.
The terminal arm of angle U lies in Quadrant 3.
The reference angle is: tan؊1 a b ‫. . .5083.0 ؍‬
2
5
So, U ‫. . .5083.0 ؉ ␲ ؍‬
‫. . .0225.3 ؍‬
The angle between ؊2␲ and 0 that is coterminal with 3.5220. . . is:
؊2␲ ؉ 3.5220. . . ‫. . .0167.2؊ ؍‬
The possible values of U are approximately: 3.5 and ؊2.7
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11. a) Convert each angle to radians.
i) 450° ii) Ϫ115°
␲
‫ 511 ؍‬a؊ b , or ؊
␲
‫ 054 ؍‬a b , or
5␲ 23␲
180 2 180 36
b) Convert each angle to degrees. Give the exact answer where
possible. Write the answer to the nearest degree where necessary.
7␲ 4␲
i) 4 ii) Ϫ 7
7 4
‫ ,)°081( ؍‬or 315° ‫081؊( ؍‬º), or approximately ؊103°
4 7
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?
arc length
Measure of central angle ‫؍‬
radius
4.5
‫ , ؍‬or 0.45
10
The area of the sector is proportional to the central angle.
Area of sector 0.45
‫؍‬
Area of circle 2␲
0.45␲(10)2
Area of sector ‫؍‬ cm2
2␲
‫ 5.22 ؍‬cm2
22 Chapter 6: Trigonometry—Checkpoint—Solutions DO NOT COPY. ©P