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1.04 Trigonometric functions

Worksheet
Sine and cosine
1

Consider the equation y = \sin x.

a

Complete the table with values in exact form:

x0\dfrac{\pi}{6}\dfrac{\pi}{2}\dfrac{5 \pi}{6}\pi\dfrac{7 \pi}{6}\dfrac{3 \pi}{2}\dfrac{11 \pi}{6}2 \pi
\sin x
b

Sketch a graph for y = \sin x on the domain -2\pi \leq 0 \leq 2\pi.

c

State the value of \sin \left(-2 \pi\right).

d

State the sign of \sin \left( \dfrac{- \pi}{12} \right).

e

State the sign of \sin \dfrac{13 \pi}{12}.

f

Which quadrant of a unit circle does an angle with measure \dfrac{13 \pi}{12} lie in?

2

Consider the equation y = \cos x.

a

Complete the table with values in exact form:

x0\dfrac{\pi}{3}\dfrac{\pi}{2}\dfrac{2 \pi}{3}\pi\dfrac{4 \pi}{3}\dfrac{3 \pi}{2}\dfrac{5 \pi}{3}2 \pi
\cos x
b

Sketch a graph for y = \cos x on the domain -2\pi \leq 0 \leq 2\pi.

c

State the value of \cos \pi.

d

State the sign of \cos \left( \dfrac{- \pi}{4} \right).

e

State the sign of \cos \dfrac{11 \pi}{6}.

f

Which quadrant of a unit circle does an angle with measure \dfrac{11 \pi}{6} lie in?

3

Consider the graph of y = \sin x given below:

\frac{1}{6}π
\frac{1}{3}π
\frac{1}{2}π
\frac{2}{3}π
\frac{5}{6}π
\frac{7}{6}π
\frac{4}{3}π
\frac{3}{2}π
\frac{5}{3}π
\frac{11}{6}π
x
-1
1
y
a

Using the graph, what is the sign of \sin \dfrac{13 \pi}{12}?

b

Which quadrant does the angle \dfrac{13 \pi}{12} lie in?

4

Consider the following unit circle:

a

State the range of y = \cos x.

b

State the range of y = \sin x.

c

How often does the graph of y = \cos x repeat?

d

How often does the graph of y = \sin x repeat?

-1
1
x
-1
1
y
5

Consider the curve y = \sin x drawn below:

-2π
-\frac{3}{2}π
-1π
-\frac{1}{2}π
\frac{1}{2}π
\frac{3}{2}π
x
-1
1
y
a

If one cycle of the graph of y = \sin x starts at x = 0, when does the next cycle start?

b

List the regions on the graph that y = \sin x is decreasing.

c

State the x-intercept in the region 0 < x < 2 \pi.

6

Consider the curve y = \cos x drawn below:

-\frac{3}{2}π
-1π
-\frac{1}{2}π
\frac{1}{2}π
\frac{3}{2}π
x
-1
1
y
a

What are the x-intercepts in the region - 2 \pi < x < 0?

b

As x approaches infinity, what y-values does the graph of y = \cos x stay between?

c

List the regions on the graph that y = \cos x is increasing.

7

Consider the functions y = \sin x and y = \cos x.

a

State the amplitude of both the graphs of these functions.

b

State the period of both the graphs of these functions.

Transformations of sine and cosine
8

Consider the graphs of \\ y = \cos x and y = \cos x + 2 shown:

Describe how to transform the graph of \\ y = \cos x to get y = \cos x + 2.

\frac{1}{2}π
x
-1
1
2
3
y
9

The function y = \cos x + 5 is translated 4 units up.

a

Write down the equation of the new function after the translation.

b

What is the maximum value of the new function?

10

Which of the following functions has a different amplitude to y = \cos x?

A

y = \cos 3 x

B

y = \cos \left( x - 3 \right)

C

y = \cos x + 3

D

y = 3 \cos x

11

Determine the equation of the graphed function given that it is of the form \\ y = a \sin x or y = a \cos x.

\frac{1}{2}π
\frac{3}{2}π
x
-3
-2
-1
1
2
3
y
12

Determine the equation of the graphed function given that it is of the form \\ y = \sin b x or y = \cos b x, where b is positive.

\frac{1}{2}π
\frac{3}{2}π
\frac{5}{2}π
\frac{7}{2}π
x
-1
-0.5
0.5
1
y
13

Consider the graph of y = \sin x:

At which value of x in the given domain would y = - \sin x have a maximum value?

\frac{1}{2}π
\frac{3}{2}π
x
-1
1
y
14

Consider the function y = - 3 \cos x.

a

State the maximum value of the function.

b

State the minimum value of the function.

c

State the amplitude of the function.

d

State the two transformations that are required to turn the graph of y = \cos x into the graph of y = - 3 \cos x.

15

Determine whether f\left(x\right)=\sin 2 x is an odd function, even function, or neither.

16

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.

-\frac{3}{4}π
-\frac{1}{2}π
-\frac{1}{4}π
\frac{1}{4}π
\frac{1}{2}π
\frac{3}{4}π
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.

17

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

c

y = 5 \sin x

d

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

e

y = \sin x + 5

18

Consider the function y = - 5 \cos x.

a

State the amplitude of the function.

b

Graph the function for 0 \leq x \leq 2\pi.

19

Sketch the graph of the function y = 2 + \sin x for 0 \leq x \leq 2\pi.

20

Consider the functions f \left( x \right) = \cos x and g \left( x \right) = \cos \left(\dfrac{x}{3}\right).

a

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

b

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

c

What transformation of the graph of f \left( x \right) results in the graph of g \left( x \right)?

d

Graph y = f \left( x \right) and y = g \left( x \right) on the same number plane for 0 \leq x \leq 2\pi.

e

Is the amplitude of g \left( x \right) different to the amplitude of f \left( x \right)?

21

Consider the function y = 4 \sin x.

a

State the amplitude of the function.

b

Graph the function for 0 \leq x \leq 2\pi.

22

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

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

c

What transformation of the graph of f \left( x \right) results in the graph of g \left( x \right)?

d

Graph y = f(x) and y = g \left( x \right) on the number plane for 0 \leq x \leq 2\pi.

23

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).

x0\dfrac{\pi}{2}\pi\dfrac{3\pi}{2}2\pi
\sin x010-10
x
\sin\left(\dfrac{x}{4}\right)010-10
24

Complete the following sentence:

The graph of the sine function crosses the x-axis for all numbers of the form , where n is an integer.

25

Consider the given graph of \\ y = \cos \left(x + \dfrac{\pi}{2}\right):

a

State the amplitude of the function.

b

Describe how the graph of y = \cos x can be transformed into the graph of \\ y = \cos \left(x + \dfrac{\pi}{2}\right).

\frac{1}{2}π
\frac{3}{2}π
x
-1
1
y
26

Determine the equation of the graphed function given that it is of the form \\ y = \sin \left(x - c\right), where c is the least positive value possible.

\frac{1}{2}π
\frac{3}{2}π
x
-1
1
y
27

Determine the values of c in the region - 2 \pi \leq c \leq 2 \pi that make: y = \sin \left(x - c\right) the same as y = \cos x.

28

What two transformations could be used to turn the graph of y = \cos x into the graph of \\ y = - \cos x + 3?

29

Describe the three transformations required to turn the graph of y = \cos x into the graph of y = - 5 \cos \left( 4 x\right).

30

Consider the function y = \cos x and the following graph:

a

Describe the transformations required to turn the graph of y = \cos x into the given graph.

b

Write the equation for the given graph?

\frac{1}{2}π
\frac{3}{2}π
x
1
2
3
4
5
y
31

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

a

Find the period of the function in radians.

b

Within the domain 0 \leq x \leq 4 \pi, what are the x-intercepts of y = 3 \sin \left(\dfrac{x}{2}\right)?

c

For 0 \leq x \leq 4 \pi, the function has a maximum value of 3. Determine the value of x at which the maximum value occurs in this domain.

32

The functions f \left( x \right) and g \left( x \right) = af \left( \dfrac{x}{b} \right) have been graphed as shown:

a

Describe the transformations that occurred on f \left( x \right) to get g \left( x \right).

b

Determine the value of a.

c

Determine the value of b.

\frac{1}{4}π
\frac{1}{2}π
\frac{3}{4}π
\frac{5}{4}π
\frac{3}{2}π
\frac{7}{4}π
x
-1
-\frac{2}{3}
-\frac{1}{3}
\frac{1}{3}
\frac{2}{3}
1
y
33

The functions f \left( x \right) and \\ g \left( x \right) = f \left( x - c \right) - d have been graphed as shown:

a

Describe the transformations that occurred on f \left( x \right) to get g \left( x \right).

b

Determine the value of d.

c

Determine the smallest positive value of c.

\frac{1}{3}π
\frac{2}{3}π
\frac{4}{3}π
\frac{5}{3}π
x
-4
-3
-2
-1
1
y
34

Consider the graphs of y = \cos x and \\ y = 3 \cos \left(x - \dfrac{\pi}{4}\right):

a

List the type of transformations that have occurred on y = \cos x to get \\ y = 3 \cos \left(x - \dfrac{\pi}{4}\right).

b

Describe how the amplitude of y = \cos x changed.

c

What phase shift has y = \cos x undergone to get y = 3 \cos \left(x - \dfrac{\pi}{4}\right)?

\frac{1}{4}π
\frac{1}{2}π
\frac{3}{4}π
\frac{5}{4}π
\frac{3}{2}π
\frac{7}{4}π
x
-3
-2
-1
1
2
3
y
35

Consider the graph of y = \sin x below. Its first maximum point for x \geq 0 is at \left(\dfrac{\pi}{2}, 1\right).

By considering the transformation that has taken place, state the coordinates of the first maximum point of each of the following functions for x \geq 0:

a

y = 5 \sin x

b

y = - 5 \sin x

c

y = \sin x + 2

d

y = \sin 3 x

e

y = \sin \left(x - \dfrac{\pi}{4}\right)

f

y = 5 \sin x + 2

\frac{1}{2}π
\frac{3}{2}π
x
-1
1
y
36

Consider the graph of y = \cos x below. Its first maximum point for x \geq 0 is at \left(0, 1\right).

By considering the transformation that has taken place, state the coordinates of the first maximum point of each of the following functions for x \geq 0:

a

y = \cos \left(x + \dfrac{\pi}{3}\right)

b

y = 5 \cos \left(x - \dfrac{\pi}{3}\right)

c

y = 2 - 5 \cos x

d

y = \cos \left(\dfrac{x}{4}\right)

e

y = 5 \cos 4 x - 2

f

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

\frac{1}{2}π
\frac{3}{2}π
x
-1
1
y
37

The graph of y = \cos x and another function that is a result of certain transformations on \\ y = \cos x is shown below:

\frac{1}{3}π
\frac{2}{3}π
\frac{4}{3}π
\frac{5}{3}π
x
-1
1
y
a

List the type of transformations that have occurred.

b

Complete the following statement:

The graph of y = \cos x has decreased its period by a factor of and then has undergone a phase shift of to the left.

c

Find the equation of the transformed graph.

38

The graph of y = \cos x undergoes the series of transformations in the following order:

  • The graph is reflected across the x-axis.

  • The graph is then horizontally translated to the left by \dfrac{\pi}{6} radians.

  • The graph is then vertically translated upwards by 5 units.

Find the equation of the transformed graph in the form y = - \cos \left(x + c\right) + d where c is the lowest positive value in radians.

39

For each of the trigonometric functions below:

i

State the period in radians.

ii
State the amplitude.
iii
State the maximum value.
iv
State the minimum value.
v
Graph the function for 0 \leq x \leq 2 \pi.
a
y = \cos 5 x
b
y = \sin \pi x
c
y = - \cos 3 x
d
y = \cos \left(x - \dfrac{\pi}{2}\right)
e
y = 5 \cos \dfrac{1}{2} x
f
y = 2 \sin 3 x
g
y = - 5 \sin 3 x
h
y = 3 \sin x + 2
i
y = \sin 2 x + 2
j
y = 2 \sin 3 x - 2
k
y = 4 - 3 \sin x
l
y = \cos 3 x + 2
m
y = 3 \sin \left(x - \dfrac{\pi}{3}\right) + 2
n
y = 2 \cos \left(x - \dfrac{\pi}{2}\right) + 3
40

For each of the following functions:

i

State the period.

ii

State the amplitude.

iii

Write down the phase shift of the function in radians.

iv

Graph the function for -\pi \leq x \leq \pi.

a
y = 4 \sin \left(x - \pi\right)
b
y = \sin \left( 2 x - \dfrac{2 \pi}{3}\right)
c
y = - 3 \sin \left( 4 x - \pi\right)
41

Consider the functions y = 2 \cos \left(x - \dfrac{\pi}{3}\right) and y = \sin \left(\dfrac{x}{4}\right).

a

Graph the two functions on the same set of axes for 0 \leq x \leq 2\pi.

b

Hence, determine the number of solutions of the equation 2 \cos \left(x - \dfrac{\pi}{3}\right) - \sin \left(\dfrac{x}{4}\right) = 0 for 0 \leq x \leq 4 \pi.

Tangent function
42

Consider the equation y = \tan x.

a

Complete the table with values in exact form:

x0\dfrac{\pi}{4}\dfrac{\pi}{2}\dfrac{3 \pi}{4}\pi\dfrac{5 \pi}{4}\dfrac{3 \pi}{2}\dfrac{7 \pi}{4}2 \pi
\tan x
b

Sketch the graph of y = \tan x on the domain -2\pi \leq 0 \leq 2\pi.

c

Graph the line y = 1 on the same coordinate plane.

d

Hence, state the exact solutions to the equation \tan x = 1 over this domain.

e

State the value of \tan \left(-2 \pi\right).

f

State the sign of \tan \left( \dfrac{- \pi}{6} \right).

g

State the sign of \tan \dfrac{9 \pi}{5}.

h

Which quadrant of a unit circle does an angle with measure \dfrac{9 \pi}{5} lie in?

43

Consider the graph of y = \tan x shown:

a

State the sign of \tan \dfrac{9 \pi}{5}.

b

Which quadrant does the angle \dfrac{9 \pi}{5} lie in?

\frac{1}{5}π
\frac{2}{5}π
\frac{3}{5}π
\frac{4}{5}π
\frac{6}{5}π
\frac{7}{5}π
\frac{8}{5}π
\frac{9}{5}π
x
-1
1
y
44

Consider the unit circle shown:

a

Express \tan \theta in terms of \sin \theta and \cos \theta.

b

Does the graph of y = \tan x repeat in regular intervals? Explain your answer.

-1
1
x
-1
1
y
45

Consider the function y = \tan \theta.

a

\tan \theta is defined as \dfrac{\text{opposite }}{\text{adjacent }} for 0 \leq \theta < \dfrac{\pi}{2} in a right-angled triangle.

What happens to the value of \tan \theta as \theta increases from 0 to \dfrac{\pi}{2}?

b

The graph of y = \cos x for 0 \leq x \leq 2 \pi is provided. For what values of x is \cos x = 0?

\frac{1}{2}π
\frac{3}{2}π
x
-1
1
y
c

Hence, for what values of x between 0 and 2 \pi is \tan x undefined?

d

Complete the table below:

x0\dfrac{\pi}{4}\dfrac{3 \pi}{4}\pi\dfrac{5 \pi}{4}\dfrac{7 \pi}{4} 2 \pi
\tan x
e

Sketch that graph of y = \tan x for 0 \leq x \leq 2 \pi.

f

Which of the following terms describes the graph?

A

Periodic

B

Decreasing

C

Even

D

Linear

g

Which of the following terms is not an appropriate description of the graph of y = \tan x?

A

Amplitude

B

Range

C

Period

D

Asymptotes

h

State the period of y = \tan x in radians.

i

State the range of y = \tan x.

j

As x increases, what would be the next asymptote of the graph after x = \dfrac{7 \pi}{2}?

Transformations of the tangent function
46

Consider the function f \left( x \right) = \tan x graphed and the function g \left( x \right) = \tan \left(x - \dfrac{\pi}{3}\right).

\frac{1}{4}π
\frac{1}{2}π
\frac{3}{4}π
\frac{5}{4}π
\frac{3}{2}π
\frac{7}{4}π
x
-2
-1
1
2
y
a

The point A on the graph of f \left( x \right) has the coordinates \left(0, 0\right).

What are the coordinates of the corresponding point on the graph of g \left( x \right)?

b

The point B on the graph of f \left( x \right) has the coordinates \left(\dfrac{\pi}{4}, 1\right).

What are the coordinates of the corresponding point on the graph of g \left( x \right)?

c

The graph of f \left( x \right) has an asymptote passing through point C with coordinates \left(\dfrac{\pi}{2}, 0\right).

What are the coordinates of the corresponding point on the graph of g \left( x \right)?

d

Hence, apply a phase shift to the graph of f \left( x \right) = \tan x to sketch the graph of \\ g \left( x \right) = \tan \left(x - \dfrac{\pi}{3}\right) for 0 \leq x \leq 2 \pi.

47

On the same set of axes, sketch the graph of y = \tan x and y = \dfrac{1}{2} \tan x for -2 \pi \leq x \leq 2 \pi.

48

On the same set of axes, sketch the graphs of the functions f \left( x \right) = - \dfrac{1}{2} \tan x and \\g \left( x \right) = 2 \tan x, on the domain -\pi \leq x \leq \pi.

49

Consider the function y = - 4 \tan \dfrac{1}{5} \left(x + \dfrac{\pi}{4}\right).

a

Find the period of the function, giving your answer in radians.

b

Find the phase shift of the function, giving your answer in radians.

c

State the range of the function.

50

Consider the function y = 6 - 3 \tan \left(x + \dfrac{\pi}{3}\right).

a

Find the period of the function, giving your answer in radians.

b

Find the phase shift of the function, giving your answer in radians.

c

State the range of the function.

51

How has the graph y = \tan \left( 2 x - \dfrac{\pi}{4}\right) been transformed from y = \tan x?

52

For each of the functions below:

i

Find the y-intercept.

ii

Find the value of y when x = \dfrac{\pi}{4}.

iii

Find the period of the function.

iv

Find the distance between the asymptotes of the function.

v

State the equation of the first asymptote of the function for x \geq 0.

vi

State the equation of the first asymptote of the function for x \leq 0.

vii

Sketch a graph of the function for -2 \pi \leq x \leq 2 \pi.

a
y = \tan x - 2
b
y = 5 \tan x + 3
53

For each of the following functions:

i

Find the y-intercept.

ii

Find the period of the function in radians.

iii

Find the distance between the asymptotes of the function.

iv

State the first asymptote of the function for x \geq 0

v

State the first asymptote of the function for x \leq 0

vi

Sketch a graph of the function for -\pi \leq x \leq \pi.

a
y = \tan 2 x
b
y = \tan 3 x
c
y = \tan \left(\dfrac{x}{2}\right)
d
y = \tan \left( - 4 x \right)
54

Consider the equation y = \tan 9 x.

a

State the period of the function in radians.

b

Sketch the graph of the function y = \tan 9 x for 0 \leq x \leq \pi.

55

Consider the function y = \tan 7 x.

a

Complete the given table of values for the function.

b

Graph the function for -\dfrac{5\pi}{28}\leq x \leq \dfrac{5\pi}{28}.

x-\dfrac{\pi}{28}0\dfrac{\pi}{28}\dfrac{3\pi}{28}\dfrac{\pi}{7}\dfrac{5\pi}{28}
y
56

A function in the form f \left( x \right) = \tan b x has adjacent x-intercepts at x = \dfrac{13 \pi}{6} and x = \dfrac{9 \pi}{4}.

a

State the equation of the asymptote lying between the two x-intercepts.

b

Find the period of the function.

c

State the equation of the function.

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