Trigonometric Graphs

Lesson

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 $\cos$`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 diagram above, we see that $\cos\frac{2\pi}{3}=-\frac{1}{2}$`c``o``s`2π3=−12 and $\sin\frac{2\pi}{3}=\frac{\sqrt{3}}{2}$`s``i``n`2π3=√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 $\cos$`c``o``s` and $\sin$`s``i``n` functions repeat the values from the angle $2\pi$2π smaller. We say $\sin$`s``i``n` and $\cos$`c``o``s` 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 $\sin x$`s``i``n``x`.

$...,x-4\pi,x-2\pi,x,x+2\pi,x+4\pi,x+6\pi,...$...,`x`−4π,`x`−2π,`x`,`x`+2π,`x`+4π,`x`+6π,...

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`′=`k``x`, we know that $\sin x'$`s``i``n``x`′ has period $2\pi$2π, but we see that $x'$`x`′ reaches $2\pi$2π when $x=\frac{2\pi}{k}$`x`=2π`k`. So, $\sin kx$`s``i``n``k``x` and $\cos kx$`c``o``s``k``x` must have period $\frac{2\pi}{k}$2π`k` with respect to $x$`x`.

The function $\sin2x$`s``i``n`2`x` begins to repeat when $2x=2\pi$2`x`=2π. That is, when $x=\pi$`x`=π. So, $\sin2x$`s``i``n`2`x` 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 $\sin kx$`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 $2\pi$2π but the period with respect to $x$`x` is $\frac{2\pi}{k}$2π`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 $3.2$3.2.

We know that $\cos kx$`c``o``s``k``x` has period $\frac{2\pi}{k}$2π`k` and, in this case, $\frac{2\pi}{k}=3.2$2π`k`=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 $\cos\left(\frac{5\pi}{8}x\right)$`c``o``s`(5π8`x`).

Consider the functions $f\left(x\right)=\sin x$`f`(`x`)=`s``i``n``x` and $g\left(x\right)=\sin3x$`g`(`x`)=`s``i``n`3`x`.

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

`f`(`x`) in radians.Complete the table of values for $g\left(x\right)$

`g`(`x`).$x$ `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 $g\left(x\right)$ `g`(`x`)$\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ State the period of $g\left(x\right)$

`g`(`x`) in radians.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

AVertical dilation by a factor of $3$3

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

CHorizontal dilation by a factor of $3$3

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

AVertical dilation by a factor of $3$3

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

CHorizontal dilation by a factor of $3$3

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 $f\left(x\right)=\cos5x$`f`(`x`)=`c``o``s`5`x`.

Determine the period of the function in radians.

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

What is the maximum value of the function?

What is the minimum value of the function?

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

`x`≤45π.Loading Graph...

Determine the equation of the graphed function given that it is of the form $y=\sin bx$`y`=`s``i``n``b``x` or $y=\cos bx$`y`=`c``o``s``b``x`, where $b$`b` is positive.

Loading Graph...

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