Grade 11

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 counterclockwise 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.

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

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

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

$x$`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°
$g\left(x\right)$`g`(`x`)	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$

State the period of $g\left(x\right)$g(x)in degrees.
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
The graph of $f\left(x\right)$f(x) has been provided below.

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

Loading Graph...

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).

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