Circular functions
Test 4
Multiple-choice questions - Calculator active
1 An observer 65 metres above sea level observes a boat out at sea at an angle of
depression of 31°. The distance of the boat from the observer is given by:
65
A
tan 59
65
B
cos59
65
C
sin59
65
D
tan 31
65
E
cos 31
2 The amplitude and period of the graph of y = −3cosπ x are:
A amplitude –3, period 2
B amplitude –3, period π
C amplitude 3, period 2
D amplitude 3, period π
E amplitude 3, period 2π
2 2
3 If sin θ = − , then a possible value of cosθ is:
3
1
A −
3
2 2
B
3
5
C −
9
1
D −
8
1
E
9
MathsWorld Mathematical Methods Units 1&2 Teacher edition
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4 The sum of the solutions to the equation cos2x = 0.5 , if 0 ≤ x ≤ 2π , is:
A π
B 2π
C 3π
D 4π
E 5π
π sin(π + x ) cos(π − x)
5 If 0 < x < , then the expression − is equivalent to:
2 cos(2π + x ) sin(2π − x)
A 0
1
B tan x +
tan x
1
C tan x −
tan x
1
D − tan x +
tan x
1
E − tan x −
tan x
[5 × 2 = 10 marks]
Short-answer questions - No Calculator
6 The graph below has equation y = acosbx + c, where a and c are integers.
a Write down the value of
i a ii c
b If b is a positive integer, find its value, and hence state the period.
[2 + 2 = 4 marks]
MathsWorld Mathematical Methods Units 1&2 Teacher edition
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3 3π
7 If cosθ = and < θ < 2π , find the exact values of
5 2
a sin θ
b tan(π − θ )
sin θ
c
tanθ
[2 + 2 + 1 = 5 marks]
8 Solve the equation 1 – 2cosπx = 2, 0 < x < 4, giving answers as exact values.
[4 marks]
9 Consider the graph of y = 2tan x + 2.
a Sketch the graph on the interval [–π, π].
b Find the exact coordinates of all axes intercepts.
[2 + 2 = 4 marks]
Extended response - Calculator Active
10 The population of lyrebirds in a Victorian national park is observed for one year.
At time t weeks after observation begins, the number of lyrebirds N can be
πt
modelled by the equation N = asin + c . The initial population of lyrebirds is
26
300, and during the first part of the observational period, the population falls to a
minimum of 210 before starting to rise again.
a Write down the value of c.
b Show that a = –90.
c Find the maximum number of lyrebirds, and state the time at which it occurs.
d Sketch the graph of N against t over the 52 weeks of the year.
e For what length of time is the number of lyrebirds 350 or more? Give your
answer correct to the nearest day.
f Find the average rate of increase in the lyrebird population over the period
during which the population rises from its minimum number to its maximum
number.
[1 + 2 + 2 + 3 + 3 + 2 = 13 marks]
MathsWorld Mathematical Methods Units 1&2 Teacher edition
Copyright © Macmillan Education Australia. Unauthorised copying prohibited.
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Answers
1 B
2 C
3 A
4 D
5 E
2π
6 a i –1 ii –2 b 3; period:
3
4 4 3
7 a − b c
5 3 5
2 4 8 10
8 , , ,
3 3 3 3
π
9 a Graph is shown below; endpoints: (−π , 2) , (π , 2) ; asymptotes: x = ± .
2
π 3π
b (− , 0), ( , 0), (0, 2)
4 4
10 a 300 b n/a c 390; at 39 weeks
d Graph is shown below; endpoints (0, 300), (52, 300).
e 16 weeks and 2 days f 6.9 lyrebirds/week
MathsWorld Mathematical Methods Units 1&2 Teacher edition
Copyright © Macmillan Education Australia. Unauthorised copying prohibited
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