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5.05 Circular functions: sine and cosine

Worksheet
Graph of sin x
1

Consider the trigonometric ratio \sin x.

a

Given the values of \sin x for angles in the first quadrant, complete the following table of values:

x0 \degree30 \degree45 \degree60 \degree90 \degree120 \degree135 \degree150 \degree180 \degree
\sin x00.50.710.8710
x210 \degree225 \degree240 \degree270 \degree300 \degree315 \degree330 \degree360 \degree
\sin x-10
b

Hence sketch the graph of y = \sin x for 0\degree \leq x \ \leq 360 \degree.

c

State the coordinates of the y-intercept.

d

State the range of the y-values.

2

Consider the function y = \sin x.

a

Complete the table, writing the values of \sin x 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 the graph for y = \sin x for -2\pi \leq x \leq 2\pi.

c

Hence state the sign of the following ratios:

i
\sin \dfrac{13 \pi}{12}
ii
\sin \dfrac{4 \pi}{3}
iii
\sin \left(-\dfrac{ \pi}{12} \right)
iv
\sin \left(-\dfrac{17 \pi}{9} \right)
d

In which quadrant of a unit circle do the following angles lie?

i
\dfrac{13 \pi}{12}
ii
\dfrac{4 \pi}{3}
iii
-\dfrac{ \pi}{12}
iv
-\dfrac{17 \pi}{9}
3

Use the diagram of the unit circle to explain the following properties of the graph of \\y=\sin x:

a

The range of values for y=\sin x is \\ -1 \leq y \leq 1.

b

The graph of y=\sin x repeats after every 2\pi radians.

-1
1
x
-1
1
y
4

Consider the graph of y = \sin x:

a

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

b

Determine whether the graph of \\ y = \sin x is increasing or decreasing on the following domains:

i

- \dfrac{\pi}{2} < x < \dfrac{\pi}{2}

ii

\dfrac{\pi}{2} < x < \dfrac{3 \pi}{2}

iii

- \dfrac{5 \pi}{2} < x < - \dfrac{3 \pi}{2}

iv

- \dfrac{3 \pi}{2} < x < - \dfrac{\pi}{2}

-2\pi
-\frac{3}{2}\pi
-1\pi
-\frac{1}{2}\pi
\frac{1}{2}\pi
1\pi
\frac{3}{2}\pi
2\pi
x
-1
1
y
c

State the x-intercept on the domain 0 < x < 2 \pi.

5

Consider the curve y = \sin x:

a

State the x-intercept on the domain - 2 \pi < x < 0.

b

If one cycle of the graph of y = \sin x starts at x = -2\pi, at what value of x does the next cycle start?

c

Determine whether the graph of \\ y = \sin x is increasing or decreasing on the following domains:

i

\dfrac{\pi}{2} < x < \dfrac{3 \pi}{2}

ii

\dfrac{3 \pi}{2} < x < \dfrac{5 \pi}{2}

iii

- \dfrac{3 \pi}{2} < x < - \dfrac{\pi}{2}

iv

- \dfrac{5 \pi}{2} < x < - \dfrac{3 \pi}{2}

-2\pi
-\frac{3}{2}\pi
-1\pi
-\frac{1}{2}\pi
\frac{1}{2}\pi
1\pi
\frac{3}{2}\pi
2\pi
x
-1
1
y
6

Consider the graph of y = \sin x and determine whether the following statements are true or false:

a

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

b

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

c

The y-values of the graph repeat after a period of 2 \pi.

-2\pi
-\frac{3}{2}\pi
-1\pi
-\frac{1}{2}\pi
\frac{1}{2}\pi
1\pi
\frac{3}{2}\pi
2\pi
x
-1
1
y
7

Consider the graph of y = \sin x.

Write an expression in terms of n to describe the x-intercepts of the function, where the angle x is measured in radians and n is an integer.

Graph of cos x
8

Consider the equation y = \cos x.

a

Complete the table, writing the values of y = \cos x 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 the graph for y = \cos x for -2\pi \leq x \leq 2\pi.

c

Hence state the sign of the following ratios:

i
\cos \dfrac{11 \pi}{6}
ii
\cos \dfrac{2 \pi}{3}
iii
\cos \left(-\dfrac{ 7\pi}{12} \right)
iv
\cos \left(\dfrac{ 10\pi}{9} \right)
d

In which quadrant of a unit circle do the following angles lie?

i
\dfrac{11 \pi}{6}
ii
\dfrac{2 \pi}{3}
iii
-\dfrac{ 7\pi}{12}
iv
\dfrac{10 \pi}{9}
9

Consider the graph of y=\cos x:

a

State the coordinates of the y-intercept.

b

State the range of the function.

c

State the period of the function.

d

Determine the x-intercepts on the domain 0 < x < 2 \pi.

-2\pi
-\frac{3}{2}\pi
-1\pi
-\frac{1}{2}\pi
\frac{1}{2}\pi
1\pi
\frac{3}{2}\pi
2\pi
x
-1
1
y
10

Consider the graph of y = \cos x:

a

If one cycle of the graph of y = \cos x starts at x = - \dfrac{\pi}{2}, at what value of x does the next cycle start?

b

Determine whether the graph of \\ y = \cos x is increasing or decreasing on the following regions:

i

- 2 \pi < x < - \pi

ii

- \pi < x < 0

iii

0 < x < \pi

iv

\pi < x < 2 \pi

-2\pi
-\frac{3}{2}\pi
-1\pi
-\frac{1}{2}\pi
\frac{1}{2}\pi
1\pi
\frac{3}{2}\pi
2\pi
x
-1
1
y
11

Consider the graph of y = \cos x:

a

State the x-intercepts on the domain - 2 \pi < x < 0.

b

If one cycle of the graph of y = \cos x starts at x = -\dfrac{3\pi}{2}, at what value of x does the next cycle start?

c

Determine whether the graph of \\ y = \cos x is increasing or decreasing on the following domains:

i

0 < x < \pi

ii

- \pi < x < 0

iii

- 2 \pi < x < - \pi

iv

\pi < x < 2 \pi

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

Consider the graph of y = \cos x and determine whether the following statements are true or false:

a

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

b

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

c

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

-2\pi
-\frac{3}{2}\pi
-1\pi
-\frac{1}{2}\pi
\frac{1}{2}\pi
1\pi
\frac{3}{2}\pi
2\pi
x
-1
1
y
13

Consider the graph of y = \cos x.

Write an expression in terms of n to describe the x-intercepts of the function, where the angle x is measured in radians and n is an integer.

14

Consider the following graphs f(x)=\sin x and g(x)=\cos x:

-2\pi
-\frac{3}{2}\pi
-1\pi
-\frac{1}{2}\pi
\frac{1}{2}\pi
1\pi
\frac{3}{2}\pi
2\pi
x
-1
1
y
-2\pi
-\frac{3}{2}\pi
-1\pi
-\frac{1}{2}\pi
\frac{1}{2}\pi
1\pi
\frac{3}{2}\pi
2\pi
x
-1
1
y

Describe the graph of g(x) in terms of a transformation of the graph of f(x).

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Outcomes

1.2.7

understand the unit circle definition of cos⁡θ, sin⁡θ and tan⁡θ and periodicity using radians

1.2.9

recognise the graphs of y=sin⁡x,y=cos⁡x, and y=tan⁡x on extended domains

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