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iGCSE (2021 Edition)

21.03 Period of sine and cosine functions

Worksheet
Horizontal and vertical dilations
1

Consider the functions f \left( x \right) = \cos x and g \left( x \right) = \cos 4 x.

a

State the period of f \left( x \right).

b

Complete the table of values for g \left( x \right).

x022.54567.590112.5135157.5180
g(x)
c

State the period of g \left( x \right).

d

Describe the transformation required to obtain the graph of g \left( x \right) from f \left( x \right).

e

Use technology to sketch the graph of g \left( x \right) for 0 \leq x \leq 180.

2

Consider the function y = 2 \cos 3 x.

a

State the amplitude of the function.

b

Find the period of the function.

c

Use technology to sketch a graph of the function for -180 \leq x \leq 180.

3

For each of the following functions:

i

State the amplitude.

ii

Find the period.

iii

Use technology to sketch a graph of the function for 0 \leq x \leq 360.

a
y = \cos 3x
b
y = \sin \left( \dfrac{x}{2} \right)
c
y = \sin \left( \dfrac{3x}{4} \right)
d
y = \cos \left(\dfrac{x}{3} \right)
e
y = - \cos 4x
f
y = 3 \sin \left(0.25x\right)
g
y = 2 \sin 3x
h
y = -5 \sin 3x
4

Consider the function y = \sin \left( \dfrac{2x}{3} \right).

a

State the amplitude of the function.

b

Find the period of the function in degrees.

c

Use technology to sketch a graph of the function for 0\degree \leq x \leq 720 \degree.

5

A table of values for the the first period of the graph y=\sin x for x \geq 0 is given in the first table on the right:

a

Complete the second table given with equivalent values for x in the the first period of the graph y = \sin \left(\dfrac{x}{4}\right) for \\x \geq 0.

b

Hence, state the period of y = \sin \left(\dfrac{x}{4}\right).

x090180270360
\sin x010-10
x
\sin\left(\dfrac{x}{4}\right)010-10
6

The functions f \left( x \right) and g \left( x \right) = f \left( kx \right) have been graphed on the same set of axes below.

-135
-90
-45
45
90
135
x
-1
1
y
a

Describe the transformation required to obtain the graph of g\left(x\right) from the graph of f \left( x \right).

b

Find the value of k.

7

For each of the following graphs, find the equation of the function given that it is of the form y = a \sin b x or y = a \cos b x, where b is positive:

a
-270
-180
-90
90
180
270
x
-3
-2
-1
1
2
3
y
b
-135
-90
-45
45
90
135
x
-2
-1
1
2
y
c
-120
-60
60
120
x
-3
-2
-1
1
2
3
y
d
-270
-180
-90
90
180
270
x
-4
-3
-2
-1
1
2
3
4
y
8

State whether the following functions represent a change in the period from the function y = \sin x:

a

y = \sin \left( 5 x\right)

b

y = 5 \sin x

c

y = \sin \left( \dfrac{x}{5} \right)

d

y = \sin x + 5

Combined translations and dilations
9

Consider the function y = \cos 3 x + 2.

a

Find the period of the function.

b

State the amplitude of the function.

c

Find the maximum value of the function.

d

Find the minimum value of the function.

e

Use technology to sketch a graph of the function for 0 \leq x \leq 360.

10

For each of the following functions:

i

State the domain of the function.

ii

State the range of the function.

iii

Use technology to sketch a graph of the function for -180 \leq x \leq 180.

a

y = \sin 2 x - 2

b

y = - 5 \sin 2 x

c

y = \sin \left(\dfrac{x}{3}\right) + 5

d

y = \cos \left(\dfrac{x}{2}\right) - 3

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Outcomes

0607E3.2G

Recognition of trigonometric, f(x) = a sin(bx); a cos(bx); tan x, function types from the shape of their graphs.

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