Lesson

We define the $\cos$`c``o``s` and $\sin$`s``i``n` 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 $\cos$`c``o``s` and $\sin$`s``i``n` functions repeat the values of the coordinates from the angle $360^\circ$360° smaller. We say $\sin$`s``i``n` and $\cos$`c``o``s` 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 $\sin\alpha$`s``i``n``α`.

$...,\alpha-720^\circ,\alpha-360^\circ,\alpha,\alpha+360^\circ,\alpha+720^\circ,\alpha+1080^\circ,...$...,`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`′=`k``α`, We know that $\sin\alpha'$`s``i``n``α`′ 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, $\sin k\alpha$`s``i``n``k``α` and $\cos k\alpha$`c``o``s``k``α` must have period $\frac{360^\circ}{k}$360°`k` with respect to $\alpha$`α`.

The function $\sin2x$`s``i``n`2`x` begins to repeat when $2x=360^\circ$2`x`=360°. That is, when $x=180^\circ$`x`=180°. So, $\sin2x$`s``i``n`2`x` 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 $\sin kx$`s``i``n``k``x` and $\cos kx$`c``o``s``k``x` where $k$`k` is a constant, the period of the function with respect to $kx$`k``x` 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 $\cos k\alpha^\circ$`c``o``s``k``α`° 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 $\cos\left(1.25\alpha^\circ\right)$`c``o``s`(1.25`α`°).

Consider the functions $f\left(x\right)=\sin x$`f`(`x`)=`s``i``n``x` and $g\left(x\right)=\sin5x$`g`(`x`)=`s``i``n`5`x`.

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 enlargement by a factor of $5$5.

AHorizontal enlargement by a factor of $\frac{1}{5}$15.

BVertical enlargement by a factor of $5$5.

CVertical enlargement by a factor of $\frac{1}{5}$15.

DHorizontal enlargement by a factor of $5$5.

AHorizontal enlargement by a factor of $\frac{1}{5}$15.

BVertical enlargement by a factor of $5$5.

CVertical enlargement by a factor of $\frac{1}{5}$15.

DThe 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...

Consider the function $f\left(x\right)=\cos x$`f`(`x`)=`c``o``s``x` and $g\left(x\right)=\cos\left(\frac{x}{2}\right)$`g`(`x`)=`c``o``s`(`x`2).

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 enlargement by a factor of $2$2.

AVertical enlargement by a factor of $2$2.

BHorizontal enlargement by a factor of $\frac{1}{2}$12.

CVertical enlargement by a factor of $\frac{1}{2}$12.

DHorizontal enlargement by a factor of $2$2.

AVertical enlargement by a factor of $2$2.

BHorizontal enlargement by a factor of $\frac{1}{2}$12.

CVertical enlargement by a factor of $\frac{1}{2}$12.

DThe 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...Is the amplitude of $g\left(x\right)$

`g`(`x`) different to the amplitude of $f\left(x\right)$`f`(`x`)?No

AYes

BNo

AYes

B

Consider the function $f\left(x\right)=\sin6x$`f`(`x`)=`s``i``n`6`x`.

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°.Loading Graph...