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°,...
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.
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.
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°).
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.
Horizontal dilation by a factor of $\frac{1}{5}$15.
Vertical dilation by a factor of $5$5.
Vertical dilation by a factor of $\frac{1}{5}$15.
The graph of $f\left(x\right)$f(x) has been provided below.
By moving the points, graph $g\left(x\right)$g(x).
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.
Complete the table of values for $g\left(x\right)$g(x).
$x$x | $0$0 | $180^\circ$180° | $360^\circ$360° | $540^\circ$540° | $720^\circ$720° | $900^\circ$900° | $1080^\circ$1080° |
---|---|---|---|---|---|---|---|
$g\left(x\right)$g(x) | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ |
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 $2$2.
Vertical dilation by a factor of $2$2.
Horizontal dilation by a factor of $\frac{1}{2}$12.
Vertical dilation by a factor of $\frac{1}{2}$12.
The graph of $f\left(x\right)$f(x) has been provided below.
By moving the points, graph $g\left(x\right)$g(x).
Is the amplitude of $g\left(x\right)$g(x) different to the amplitude of $f\left(x\right)$f(x)?
No
Yes
Consider the function $f\left(x\right)=\sin6x$f(x)=sin6x.
Determine the period of the function in degrees.
How many cycles does the curve complete in $3240^\circ$3240°?
What is the maximum value of the function?
What is the minimum value of the function?
Graph the function for $0^\circ\le x\le120^\circ$0°≤x≤120°.