New Zealand
Level 7 - NCEA Level 2

# Period changes for sine and cosine

Lesson

## Preliminaries

We define the $\cos$cos and $\sin$sin functions as the horizontal and vertical coordinates of a point that moves on the unit circle. In the diagrams below, this is shown for an angle $\alpha$α in the first and second quadrants.

If we imagine the point moving anticlockwise on the unit circle so that the radius from the point makes an ever-increasing angle with the positive horizontal axis, eventually the angle exceeds $360^\circ$360°; but the values of the $\cos$cos and $\sin$sin functions repeat the values of the coordinates from the angle $360^\circ$360° smaller. We say $\sin$sin and $\cos$cos are periodic functions with period $360^\circ$360°.

Thus, for any angle $\alpha$α, there is a sequence of angles with the same value of $\sin\alpha$sinα.

$...,\alpha-720^\circ,\alpha-360^\circ,\alpha,\alpha+360^\circ,\alpha+720^\circ,\alpha+1080^\circ,...$...,α720°,α360°,α,α+360°,α+720°,α+1080°,...

## Transformations involving period

Again, consider the angle $\alpha$α made by the point moving around the unit circle. If a new angle $\alpha'$α is defined by $\alpha'=k\alpha$α=kα, We know that $\sin\alpha'$sinα has period $360^\circ$360°, but we see that $\alpha'$α reaches $360^\circ$360°  when $\alpha=\frac{360^\circ}{k}$α=360°k. So, $\sin k\alpha$sinkα and $\cos k\alpha$coskα must have period $\frac{360^\circ}{k}$360°k with respect to $\alpha$α.

#### Example 1

The function $\sin2x$sin2x begins to repeat when $2x=360^\circ$2x=360°. That is, when $x=180^\circ$x=180°. So, $\sin2x$sin2x has period $180^\circ$180°. The period is multiplied by $\frac{1}{2}$12 when $x$x is multiplied by $2$2.

Thus, we see that for functions $\sin kx$sinkx and $\cos kx$coskx where $k$k is a constant, the period of the function with respect to $kx$kx is $360^\circ$360° but the period with respect to $x$x is $\frac{360^\circ}{k}$360°k.

We can use these ideas to deduce the formula for a sine or cosine function from a graph.

#### Example 2

This graph looks like the graph of a cosine function since it has the value $1$1 at $0$0. However, the period is $288^\circ$288°.

We know that $\cos k\alpha^\circ$coskα° has period $\frac{360^\circ}{k}$360°k and, in this case, $\frac{360^\circ}{k}=288^\circ$360°k=288°. Therefore, $k=\frac{360^\circ}{288}=1.25$k=360°288=1.25.

The graph must belong to the function given by $\cos\left(1.25\alpha^\circ\right)$cos(1.25α°).

#### Worked examples

##### Question 1

Consider the functions $f\left(x\right)=\sin x$f(x)=sinx and $g\left(x\right)=\sin5x$g(x)=sin5x.

1. State the period of $f\left(x\right)$f(x) in degrees.

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

 $x$x $g\left(x\right)$g(x) $0^\circ$0° $18^\circ$18° $36^\circ$36° $54^\circ$54° $72^\circ$72° $90^\circ$90° $108^\circ$108° $126^\circ$126° $144^\circ$144° $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
3. State the period of $g\left(x\right)$g(x)in degrees.

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

Horizontal dilation by a factor of $5$5.

A

Horizontal dilation by a factor of $\frac{1}{5}$15.

B

Vertical dilation by a factor of $5$5.

C

Vertical dilation by a factor of $\frac{1}{5}$15.

D

Horizontal dilation by a factor of $5$5.

A

Horizontal dilation by a factor of $\frac{1}{5}$15.

B

Vertical dilation by a factor of $5$5.

C

Vertical dilation by a factor of $\frac{1}{5}$15.

D
5. The graph of $f\left(x\right)$f(x) has been provided below.

By moving the points, graph $g\left(x\right)$g(x).

##### Question 2

Consider the function $f\left(x\right)=\cos x$f(x)=cosx and $g\left(x\right)=\cos\left(\frac{x}{2}\right)$g(x)=cos(x2).

1. State the period of $f\left(x\right)$f(x) in degrees.

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

 $x$x $g\left(x\right)$g(x) $0$0 $180^\circ$180° $360^\circ$360° $540^\circ$540° $720^\circ$720° $900^\circ$900° $1080^\circ$1080° $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
3. State the period of $g\left(x\right)$g(x) in degrees.

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

Horizontal dilation by a factor of $2$2.

A

Vertical dilation by a factor of $2$2.

B

Horizontal dilation by a factor of $\frac{1}{2}$12.

C

Vertical dilation by a factor of $\frac{1}{2}$12.

D

Horizontal dilation by a factor of $2$2.

A

Vertical dilation by a factor of $2$2.

B

Horizontal dilation by a factor of $\frac{1}{2}$12.

C

Vertical dilation by a factor of $\frac{1}{2}$12.

D
5. The graph of $f\left(x\right)$f(x) has been provided below.

By moving the points, graph $g\left(x\right)$g(x).

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

No

A

Yes

B

No

A

Yes

B

##### Question 3

Consider the function $f\left(x\right)=\sin6x$f(x)=sin6x.

1. Determine the period of the function in degrees.

2. How many cycles does the curve complete in $3240^\circ$3240°?

3. What is the maximum value of the function?

4. What is the minimum value of the function?

5. Graph the function for $0^\circ\le x\le120^\circ$0°x120°.

### Outcomes

#### M7-2

Display the graphs of linear and non-linear functions and connect the structure of the functions with their graphs

#### 91257

Apply graphical methods in solving problems