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5.02 Graphs of trigonometric functions

Worksheet
Sine function
1

Consider the function y = \sin x.

a

Complete the table of values giving answers in exact form:

x0\degree30\degree90\degree150\degree180\degree210\degree270\degree330\degree360\degree
\sin x
b

Sketch the graph of y = \sin x for 0 \leq x \leq 360.

c

State the sign of \sin 324 \degree.

d

Which quadrant would the angle 324 \degree lie?

e

Describe the trend of the graph of y = \sin x.

f

State the period of y = \sin x.

2

Given the unit circle below, state whether the following statements are true of the graph of y = \sin \theta:

a

The range of values of y = \sin \theta is -\infty \lt y \lt \infty.

b

The values of y = \sin \theta lie in the range - 1 \leq y \leq 1.

c

The graph of y = \sin \theta repeats after every 180 \degree.

d

The graph of y = \sin \theta repeats after every 360 \degree.

-1
1
x
-1
1
y
3

Consider the function y = \sin x.

a

State the y-intercept.

b

State the maximum y-value.

c

State the minimum y-value.

d

State the range of the function.

4

Determine whether the following statements are true of the curve y = \sin x.

a

The graph of y = \sin x is cyclic.

b

As x approaches infinity, the height of the graph approaches infinity.

c

The graph of y = \sin x is increasing between x = - 360 \degree and x = - 270 \degree.

d

The graph of y = \sin x is symmetric about the line x = 0.

e

The graph of y = \sin x is symmetric with respect to the origin.

f

The y-values of the graph repeat after a period of 180 \degree.

g

As x approaches infinity, the graph of y = \sin x stays between y = - 1 and y = 1.

5

Consider the function y = \sin x.

a

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

b

Describe whether the graph is increasing or decreasing in the following intervals:

i

- 270 \degree \lt x \lt - 90 \degree

ii

- 450 \degree \lt x \lt - 270 \degree

iii

90 \degree \lt x \lt 270 \degree

iv

- 90 \degree \lt x \lt 90 \degree

c

Find the x-intercept in the following intervals:

i

0 \degree \lt x \lt 360 \degree

ii

- 360 \degree \lt x \lt 0 \degree

Cosine function
6

Consider the function y = \cos x.

a

Complete the table of values giving answers in exact form:

x0\degree60\degree90\degree120\degree180\degree240\degree270\degree300\degree360\degree
\cos x
b

Sketch the graph of y = \cos x for 0 \leq x \leq 360.

c

State the sign of \cos 340 \degree.

d

State the quadrant where an angle with measure 340 \degree lie.

7

Given the unit circle below, state whether the following statements are true of the graph of \\ y = \cos \theta:

a

The values of y = \cos \theta lie in the range \\ - 1 \leq y \leq 1.

b

The range of values of y = \cos \theta is \\ -\infty \lt y \lt \infty.

c

The graph of y = \cos \theta repeats after every 180 \degree.

d

The graph of y = \cos \theta repeats after every 360 \degree.

-1
1
x
-1
1
y
8

Consider the function y = \cos x.

a

State the y-intercept.

b

State the maximum y-value.

c

State the minimum y-value.

d

State the domain of the function.

9

State whether the following statements are true of the graph of y = \cos x:

a

The graph of y = \cos x is cyclic.

b

As x approaches infinity, the height of the graph approaches infinity.

c

The graph of y = \cos x is increasing between x = 90 \degree and x = 180 \degree.

d

The graph of y = \cos x is symmetric about the line x = 0.

e

The graph of y = \cos x is symmetric with respect to the origin.

f

The y-values of the graph repeat after a period of 180 \degree.

g

As x approaches infinity, the graph of y = \cos x stays between y = - 1 and y = 1.

10

Consider the function y = \cos x.

a

If one cycle of the graph of y = \cos x starts at x = - 90 \degree, where does the next cycle start?

b

Describe whether the graph of y = \cos x is increasing or decreasing in the following intervals:

i

- 180 \degree \lt x \lt 0 \degree

ii

- 360 \degree \lt x \lt - 180 \degree

iii

0 \degree \lt x \lt 180 \degree

iv

180 \degree \lt x \lt 360 \degree

c

Find the x-value of the x-intercepts in the following intervals:

i

0 \degree \lt x \lt 360 \degree

ii

- 360 \degree \lt x \lt 0 \degree

Tangent function
11

Consider the equation y = \tan x.

a

Complete the table of values, giving answers in exact form. Write '-' if the value is undefined:

x0\degree60\degree90\degree120\degree135\degree180\degree225\degree270\degree300\degree360\degree
\tan x
b

Sketch the graph of y = \tan x for 0 \leq x \leq 360.

c

State the y-intercept.

d

State the sign of \tan 345 \degree.

e

State the quadrant where an angle with measure 345 \degree lie.

12

Given the unit circle below, state whether the following statements are true of the graph of y = \tan x:

a

The graph of y = \tan x repeats in regular intervals since the values of \sin x and \cos x repeat in regular intervals.

b

The graph of y = \tan x is defined for all values of x.

c

Since the radius of the circle is one unit, the value of the function y = \tan x lies in the region - 1 \leq y \leq 1.

d

The range of values of function \\ y = \tan x is -\infty \lt y \lt \infty.

-1
1
x
-1
1
y
Applications
13

By considering the graphs of the relevant trigonometric functions, evaluate the following:

a

\sin 90 \degree

b

\cos 90 \degree

c

\tan 90 \degree

d

\sin 180 \degree

e

\cos 180 \degree

f

\tan 180 \degree

g

\sin 270 \degree

h

\cos 270 \degree

i

\tan 270 \degree

j

\sin 360 \degree

k

\cos 360 \degree

l

\tan 360 \degree

14

Consider the function y = \sin \theta.

Find \theta if:

a

\sin \theta = \sin 390 \degree where \theta is in the 1st quadrant.

b

\sin \theta = \sin 480 \degree where \theta is in the 2nd quadrant.

c

\sin \theta = \sin 570 \degree where \theta is in the 3rd quadrant.

15

Consider the equation \sin x = \cos x in the domain 0 \degree \leq x \leq 360 \degree.

a

Sketch the graph of y = \sin x and y = \cos x on the same coordinate axes.

b

State the number of solutions of the equation \sin x = \cos x in the domain 0 \degree \leq x \leq 360 \degree.

16

Consider the curve y = \cos x for 0 \degree \leq x \leq 360 \degree.

a

How long is one cycle of the graph?

b

State the x-values for which:

i
\cos x = 0
ii
\cos x = 0.5
iii
\cos x = - 0.5
17

Consider the curve y = \sin x for 0 \degree \leq x \leq 360 \degree.

a

How long is one cycle of the graph?

b

State the x-values for which:

i
\sin x = 0
ii
\sin x = 0.5
iii
\sin x = -0.5
18

Consider the curve y = \tan x for 0 \degree \leq x \leq 360 \degree.

a

How long is one cycle of the graph?

b

State the x-values for which:

i
\tan x = 0
ii
\tan x = 1
iii
\tan x = - 1
19

Consider the function y = \cos x for - 180 \degree \leq x \leq 180 \degree.

a

Sketch the function.

b

State the x-values for which:

i
\cos x = 0
ii
\cos x = \pm 1
iii
\cos x = \pm \dfrac{1}{2}
iv
\cos x = \pm \dfrac{1}{\sqrt{2}}
20

Consider the function y = \sin x for - 180 \degree \leq x \leq 180 \degree.

a

Sketch the function.

b

State the x-values for which:

i
\sin x = 0
ii
\sin x = \pm 1
iii
\sin x = \pm \dfrac{\sqrt{3}}{2}
iv
\sin x = \pm \dfrac{1}{\sqrt{2}}
21

Consider the function y = \tan x for - 180 \degree \leq x \leq 180 \degree.

a

Sketch the function.

b

State the x-values for which:

i
\tan x = 0
ii
\tan x = \pm 1
iii
\tan x = \pm \sqrt{3}
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Outcomes

MA11-1

uses algebraic and graphical techniques to solve, and where appropriate, compare alternative solutions to problems

MA11-3

uses the concepts and techniques of trigonometry in the solution of equations and problems involving geometric shapes

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