 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