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IB Math Studies – Chapter 16 – Trigonometric Functions – Review Questions
1.
The diagram shows the graph of y = sin ax + b.
y
2
1
0
30°
60°
x
90° 120° 150° 180° 210° 240° 270° 300° 330° 360°
(a)
Using the graph, write down the following values
(i)
the period;
(ii)
the amplitude;
(iii)
b.
(b)
Calculate the value of a.
Working:
Answers:
(a) (i) ...........................................................
(ii) ...........................................................
(iii) ...........................................................
(b) ..................................................................
(Total 8 marks)
2.
(a)
Sketch the graph of the function y =1+
sin (2 x )
2
for 0  x  360 on the
axes below.
(4)
y
2
1
90
180
270
x
360
–1
–2
(b)
Write down the period of the function.
(c)
Write down the amplitude of the function.
(1)
(1)
Working:
Answers:
(b) ...................................................
(c) ...................................................
(Total 6 marks)
1
3.
The diagram below shows the graph of y = – a sin x° + c, 0 ≤ x ≤ 360.
y
5
4
3
2
1
x
0
90
270
180
360
–1
–2
–3
Use the graph to find the values of
(a)
c;
(b)
a.
Working:
Answers:
(a) …………………………………………..
(b) ..................................................................
(Total 4 marks)
4.
The graph below shows part of the function y = 2 sin x + 3.
y
6
5
4
3
2
1
0
0º
(a)
90º
180º
270º
360º
450º
x
Write the domain of the part of the function shown on the graph.
(b)
Write the range of the part of the function shown on the graph.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 4 marks)
2
5.
The graphs of three trigonometric functions are drawn below. The x variable is measured in degrees, with 0 ≤ x ≤ 360°. The
amplitude 'a' is a positive constant with 0 < a ≤ 1.
Graph A
Graph B
a
2
y
0
90
180
x
270
360
–a
90
180
x
270
360
Graph C
a
0
90
180
270
360
–a
(a)
Write the letter of the graph next to the function representing that graph in the box below.
FUNCTION
GRAPH
y = a cos (x)
y = a sin (2x)
y = 2 + a sin (x)
(b)
State the period of the function shown in graph B.
(c)
State the range of the function 2 + a sin (x) in terms of the constant a.
Working:
Answers:
(b) ...................................................
(c) ...................................................
(Total 8 marks)
3
6.
(a)
(b)
For y = 0.5 cos 0.5 x, find
(i)
the amplitude;
(ii)
the period.
Let y = –3 sin x + 2, where 90° ≤ x ≤ 270°.
By drawing the graph of y or otherwise, complete the table below for the given values of y.
x
y
–1
2
Working:
Answers:
(a) (i) ……………………………………..
(ii) ……………………………………..
(Total 4 marks)
IB Math Studies – Chapter 16 – Trigonometric – Review Questions Mark Scheme
1.
(a)
(b)
(i)
(ii)
120°
1
(iii)
1
(A2)
(C2)
(A2)
(C2)
(A2)
(C2)
360
= 120  a = 3
a
(A2)
(C2)
[8]
2.
(a)
(A4)(C4)
Notes: (A1) for correct y-intercept
(A1) for correct minimum points
(A1) for correct maximum points
(A1) for smooth sine curve.
(b)
period = 180
(c)
amplitude =
1
2
(A1)(ft)
(C1)
(A1)(ft)
(C1)
[6]
4
3.
(a)
c=1
(A1)
(C1)
(b)
amplitude =
42
2
(M1)
=3
The graph of y = sin x° has been reflected in a line parallel
to the x-axis therefore a = –3
(A1)
(A1)
(C3)
[4]
4.
(a)
(b)
0°  x  450°
(A2)
Note:
Award (A1) for x  0°, (A1) for x  450°.
Award (A1) for 0° and 450° if the inequalities are incorrect.
Note:
Award (A1) for y  1, (A1) for y  5.
Award (A1) for 1 and 5 if the inequalities are incorrect.
Award (A2) if the candidates have the range and domain reversed, that is,
(a)
1y5
(b)
0° < x < 450°
1y5
Note:
(A2)
[4]
5.
(a)
FUNCTION
y = a cos (x)
y = a sin (2x)
y = 2 + a sin (x)
(b)
P=
GRAPH LABEL
C
B
A
(A1)(A1)(A1)
(C3)
360
= 180
2
(M1)(A1)
or the period is 180 degrees.
(C2)
Note: Award (A1) only for 0 – 180 or π.
(c)
The range is [2 – a, 2 + a] or 2 – a  y  2 + a
1)
Note: Award (A1) for each value seen and (A1) for correct [ ] or  y .
[–a, a] (or equivalent) can receive (A2) and (–a, a) or equivalent can receive (A1).
(A1)(A1)(A
(C3)
[8]
6.
(a)
(i)
(ii)
0.5
720°
(A1)
(A1)
(b)
x
90°
180°
y
–1
2
(A1)
(A1)
[4]
5