New Zealand
Level 8 - NCEA Level 3

# Period changes for sine and cosine

Lesson

## Preliminaries

We define the radian measure of an angle in terms of the length of the arc associated with the angle in the unit circle. There must be $2\pi$2π radians in a full circle because this is the length of the circumference. In the diagram above, the arc associated with the angle $\frac{2\pi}{3}$2π3 has length $\frac{2\pi}{3}$2π3.

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 diagram above, we see that $\cos\frac{2\pi}{3}=-\frac{1}{2}$cos2π3=12 and $\sin\frac{2\pi}{3}=\frac{\sqrt{3}}{2}$sin2π3=32.

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 $2\pi$2π; but the values of the $\cos$cos and $\sin$sin functions repeat the values from the angle $2\pi$2π smaller. We say $\sin$sin and $\cos$cos are periodic functions with period $2\pi$2π.

Thus, for any angle $x$x, there is a sequence of angles with the same value of $\sin x$sinx.

$...,x-4\pi,x-2\pi,x,x+2\pi,x+4\pi,x+6\pi,...$...,x4π,x2π,x,x+2π,x+4π,x+6π,...

## Transformations involving period

Again, consider the angle $x$x made by the point moving around the unit circle. If a new angle $x'$x is defined by $x'=kx$x=kx, we know that $\sin x'$sinx has period $2\pi$2π, but we see that $x'$x reaches $2\pi$2π when $x=\frac{2\pi}{k}$x=2πk. So, $\sin kx$sinkx and $\cos kx$coskx must have period $\frac{2\pi}{k}$2πk with respect to $x$x.

#### Example 1

The function $\sin2x$sin2x begins to repeat when $2x=2\pi$2x=2π. That is, when $x=\pi$x=π. So, $\sin2x$sin2x has period $\pi$π. 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 $2\pi$2π but the period with respect to $x$x is $\frac{2\pi}{k}$2πk.

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 $3.2$3.2.

We know that $\cos kx$coskx has period $\frac{2\pi}{k}$2πk and, in this case, $\frac{2\pi}{k}=3.2$2πk=3.2. Therefore, $k=\frac{2\pi}{3.2}=\frac{2\pi}{\frac{16}{5}}=\frac{5\pi}{8}$k=2π3.2=2π165=5π8.

The graph must belong to the function given by $\cos\left(\frac{5\pi}{8}x\right)$cos(5π8x).

#### Worked examples

##### Question 1

Consider the functions $f\left(x\right)=\sin x$f(x)=sinx and $g\left(x\right)=\sin3x$g(x)=sin3x.

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

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

 $x$x $g\left(x\right)$g(x) $0$0 $\frac{\pi}{6}$π6​ $\frac{\pi}{3}$π3​ $\frac{\pi}{2}$π2​ $\frac{2\pi}{3}$2π3​ $\frac{5\pi}{6}$5π6​ $\pi$π $\frac{7\pi}{6}$7π6​ $\frac{4\pi}{3}$4π3​ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
3. State the period of $g\left(x\right)$g(x) in radians.

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

Vertical dilation by a factor of $\frac{1}{3}$13

A

Vertical dilation by a factor of $3$3

B

Horizontal dilation by a factor of $\frac{1}{3}$13

C

Horizontal dilation by a factor of $3$3

D

Vertical dilation by a factor of $\frac{1}{3}$13

A

Vertical dilation by a factor of $3$3

B

Horizontal dilation by a factor of $\frac{1}{3}$13

C

Horizontal dilation by a factor of $3$3

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 $f\left(x\right)=\cos5x$f(x)=cos5x.

1. Determine the period of the function in radians.

2. How many cycles does the curve complete in $15$15 radians?

3. What is the maximum value of the function?

4. What is the minimum value of the function?

5. Graph the function for $0\le x\le\frac{4}{5}\pi$0x45π.

##### Question 3

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

### Outcomes

#### M8-2

Display and interpret the graphs of functions with the graphs of their inverse and/or reciprocal functions