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R E V I E W, p a g e s 5 6 0 – 5 6 6
6.1
1. Determine the value of each trigonometric ratio. Use exact values
where possible; otherwise write the value to the nearest thousandth.
a) tan (⫺45°)
b) cos 600°
ⴝ ⴚ1
c) sec (⫺210°)
ⴝ cos 240º
ⴝⴚ
d) sin 765°
1
2
e) cot 21°
ⴝ sin 45º
1
2
ⴝ√
1
cos ( ⴚ 210°)
1
ⴝ
cos 150°
2
ⴝ ⴚ√
3
ⴝ
f) csc 318°
1
ⴝ
tan 21°
ⴝ
⬟ 2.605
⬟ ⴚ1.494
1
sin 318°
2. To the nearest degree, determine all possible values of u for which
cos u ⫽ 0.76, when ⫺360° ≤ u ≤ 360°.
Since cos U is positive, the terminal arm of angle U lies in Quadrant 1 or 4.
The reference angle is: cosⴚ1(0.76) ⬟ 41º
For the domain 0° ◊ U ◊ 360°:
In Quadrant 1, U ⬟ 41º
In Quadrant 4, U ⬟ 360º ⴚ 41º, or approximately 319º
For the domain ⴚ360° ◊ U ◊ 0°:
In Quadrant 1, U ⬟ ⴚ360º ⴙ 41º, or approximately ⴚ319º
In Quadrant 4, U ⬟ ⴚ41º
6.2
3. As a fraction of ␲, determine the length of the arc that subtends a
central angle of 225° in a circle with radius 3 units.
Arc length:
15
225
(2␲)(3) ⴝ ␲
360
4
6.3
4. a) Convert each angle to degrees. Give the answer to the nearest
degree where necessary.
5␲
i) 3
ⴝ
5(180°)
3
ⴝ 300°
44
10␲
ii) ⫺ 7
ⴝⴚ
10(180°)
7
⬟ ⴚ257°
Chapter 6: Trigonometry—Review—Solutions
iii) 4
ⴝ 4a ␲ b
180°
⬟ 229º
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b) Convert each angle to radians.
ii) ⫺240°
i) 150°
␲
b
180
␲
b
180
ⴝ 150 a
ⴝ
iii) 485°
␲
b
180
ⴝ ⴚ240 a
5␲
6
ⴝⴚ
ⴝ 485 a
4␲
3
ⴝ
97␲
36
5. In a circle with radius 5 cm, an arc of length 6 cm subtends a central
angle. What is the measure of this angle in radians, and to the
nearest degree?
Angle measure is:
arc length
radius
ⴝ
6
5
ⴝ 1.2
180°
In degrees, 1.2 ⴝ 1.2 a ␲ b
⬟ 69º
The angle measure is 1.2 radians or approximately 69º.
6. A race car is travelling around a circular track at an average speed of
120 km/h. The track has a diameter of 1 km. Visualize a line
segment joining the race car to the centre of the track. Through
what angle, in radians, will the segment have rotated in 10 s?
1
120
km ⴝ
km
60 # 60
30
1
10
So, in 10 s, the car travels: km ⴝ km
30
3
1
arc length
3
Angle measure is:
ⴝ
1
radius
1
ⴝ
3
In 1 s, the car travels:
In 10 s, the segment will have rotated through an angle of
1
radian.
3
7. Determine the value of each trigonometric ratio. Use exact values
where possible; otherwise write the value to the nearest thousandth.
␲
5␲
a) sin 3
b) cos 6
√
ⴝ
√
3
2
ⴝⴚ
3
2
␲
c) sec a- 2 b
ⴝ
1
␲
cos aⴚ b
2
,
which is undefined
15␲
d) tan 4
ⴝ tan
3␲
4
ⴝ ⴚ1
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e) csc 5
ⴝ
1
sin 5
⬟ ⴚ1.043
f) cot (⫺22.8)
ⴝ
1
tan ( ⴚ 22.8)
⬟ ⴚ0.954
Chapter 6: Trigonometry—Review—Solutions
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8. P(3, ⫺1) is a terminal point of angle u in standard position.
a) Determine the exact values of all the trigonometric ratios for u.
Let the distance between the origin and P be r.
Substitute: x ⴝ 3, y ⴝ ⴚ1
Use: x2 ⴙ y2 ⴝ r2
9 ⴙ 1 ⴝ r2
√
r ⴝ 10
√
3
1
sin U ⴝ ⴚ √
csc U ⴝ ⴚ 10
cos U ⴝ √
10
√
10
sec U ⴝ
3
10
tan U ⴝ ⴚ
1
3
cot U ⴝ ⴚ3
b) To the nearest tenth of a radian, determine possible values of u in
the domain ⫺2␲ ≤ u ≤ 2␲.
The terminal arm of angle U lies in Quadrant 4.
The reference angle is: tanⴚ1 a b ⴝ 0.3217. . .
1
3
So, U ⴝ ⴚ0.3217. . .
The angle between 0 and 2␲ that is coterminal with ⴚ0.3217. . . is:
2␲ ⴚ 0.3217. . . ⴝ 5.9614. . .
Possible values of U are approximately: 6.0 and ⴚ0.3
6.4
9. Use graphing technology to graph each function below for
⫺2␲ ≤ x ≤ 2␲, then list these characteristics of the graph:
amplitude, period, zeros, domain, range, and the equations of the
asymptotes.
a) y ⫽ sin x
The amplitude is 1. The period is 2␲. The zeros are 0, — ␲, — 2␲. The
domain is ⴚ2␲ ◊ x ◊ 2␲. The range is ⴚ1 ◊ y ◊ 1. There are no
asymptotes.
b) y ⫽ cos x
␲
2
The amplitude is 1. The period is 2␲. The zeros are — , —
The domain is ⴚ2␲
asymptotes.
3␲
.
2
◊ x ◊ 2␲. The range is ⴚ1 ◊ y ◊ 1. There are no
c) y ⫽ tan x
There is no amplitude. The period is ␲. The zeros are 0, — ␲, — 2␲.
3␲
. The range is y ç ⺢. The equations of
2
␲
3␲
the asymptotes are x ⴝ — and x ⴝ — .
2
2
␲
2
The domain is x ⴝ — , —
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Chapter 6: Trigonometry—Review—Solutions
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6.5
10. On the same grid, sketch graphs of the functions in each pair for
0 ≤ x ≤ 2␲, then describe your strategy.
␲
a) y ⫽ sin x and y ⫽ sin ax - 4 b
1
y
y sin x
2
0
1
x
3
2
y sin x 4
(
)
For the graph of y ⴝ sin x, I used the completed table of values from
Lesson 6.4.
The horizontal scale is 1 square to
I then shifted several points
␲
␲
units, because the phase shift is .
4
4
␲
units right and joined the points to get
4
␲
4
the graph of y ⴝ sin ax ⴚ b .
3
b) y ⫽ cos x and y ⫽ 2 cos x
3
y cos x
2
y
1
0
1
y cos x
x
3
5
3
For the graph of y ⴝ cos x, I used the completed table of values from
Lesson 6.4.
I multiplied every y-coordinate by 1.5, plotted the new points, then
joined them to get the graph of y ⴝ
©P DO NOT COPY.
3
cos x.
2
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6.6
␲
1
11. a) Graph y ⫽ 2 sin 3 ax + 6 b ⫹ 2 for ⫺2␲ ≤ x ≤ 2␲.
Explain your strategy.
3
y
y
1
2
sin 3 x 6
2
(
)
2
y
1
sin 3x
2
1
x
0
11
6
2
7
6
6
1
Sample response: I graphed y ⴝ
5
6
3
2
␲
1
sin 3x, shifted several points units
2
6
left and 2 units up, then joined the points to get the graph of
yⴝ
␲
1
sin 3 ax ⴙ b ⴙ 2.
2
6
b) List the characteristics of the graph you drew.
1
2
2␲
. There are no zeros.
3
3
5
2␲. The range is ◊ y ◊ .
2
2
The amplitude is . The period is
The domain is ⴚ2␲
◊x◊
12. An equation of the function graphed below has the form
y ⫽ a cos b(x ⫺ c) ⫹ d. Identify the values of a, b, c, and d in the
equation, then write an equation for the function.
y
2
y f(x)
x
0
2
3
4
2
1
2
Sample response: The equation of the centre line is y ⴝ , so the
1
unit up and d ⴝ
2
3 ⴚ ( ⴚ 2)
5
The amplitude is:
ⴝ , so a ⴝ
2
2
vertical translation is
1
.
2
5
2
Choose the x-coordinates of two adjacent maximum points,
␲
␲
11␲
11␲
and
. The period is:
ⴚ aⴚ b ⴝ 4␲
3
3
3
3
2␲
1
So, b is:
ⴝ
4␲
2
ⴚ
To the left of the y-axis, the cosine function begins its cycle at
␲
3
␲
3
␲
3
x ⴝ ⴚ , so a possible phase shift is ⴚ , and c ⴝ ⴚ .
Substitute for a, b, c, and d in: y ⴝ a cos b(x ⴚ c) ⴙ d
An equation is: y ⴝ
48
␲
1
5
1
cos ax ⴙ b ⴙ
2
2
3
2
Chapter 6: Trigonometry—Review—Solutions
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6.7
13. A water wheel has diameter 10 m and
completes 4 revolutions each minute.
The axle of the wheel is 8 m above
a river.
10 m
a) The wheel is at rest at time t ⫽ 0 s,
with point P at the lowest point on
the wheel. Determine a function that 8 m
models the height of P above the river,
h metres, at any time t seconds.
Explain how the characteristics of
the graph relate to the given information.
P
River
The time for 1 revolution is 15 s.
h
At t ⴝ 0, h ⴝ 3
At t ⴝ 7.5, h ⴝ 13
(7.5, 13)
The graph begins at (0, 3), which is a
minimum point.
The first maximum point is at (7.5, 13).
(15, 3)
The next minimum point is after 1 cycle
(0, 3)
and it has coordinates (15, 3).
t
The position of the first maximum is
0
known, so use a cosine function:
h(t) ⴝ a cos b(t ⴚ c) ⴙ d
The constant in the equation of the centre line of the graph is the
height of the axle above the river, so its equation is: h ⴝ 8; and this is
also the vertical translation, so d ⴝ 8
The amplitude is one-half the diameter of the wheel, so a ⴝ 5
2␲
The period is the time for 1 revolution, so b ⴝ
15
A possible phase shift is: c ⴝ 7.5
2␲
An equation is: h(t) ⴝ 5 cos (t ⴚ 7.5) ⴙ 8
15
b) Use technology to graph the function. Use this graph to
determine:
i) the height of P after 35 s
Graph: Y ⴝ 5 cos
2␲
(X ⴚ 7.5) ⴙ 8
15
Determine the Y-value when X ⴝ 35.
After 35 s, P is 10.5 m high.
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ii) the times, to the nearest tenth of a second, in the first 15 s of
motion that P is 11 m above the river
Graph: Y ⴝ 5 cos
2␲
(X ⴚ 7.5) ⴙ 8 and Y ⴝ 11
15
Determine the Y-coordinates of the first two points of intersection.
P is 11 m above the river after approximately 5.3 s and 9.7 s.
50
Chapter 6: Trigonometry—Review—Solutions
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