Periodic Functions

8.2 Graphs of the Other Trigonometric Functions

Graphs of the Other Trigonometric Functions

Sine and cosine are smooth and wave-like, but the other four trig functions have a very different kind of behavior. Tangent, cotangent, secant, and cosecant all blow up to infinity at certain points, so asymptotes are now part of the picture.

Tangent and Cotangent

Instead of smooth waves, tangent and cotangent have repeating branches that fly off to $\pm\infty$ near their asymptotes. Their range is all real numbers. That is a big difference from sine and cosine.

Tangent and Cotangent

Open interactive graph in GeoGebra

Fact: Tangent and Cotangent Features

For tangent and cotangent:

\[ \begin{align*} \text{Period} &= \pi \text{ for the parent functions} \\ \text{Range} &= (-\infty,\infty) \end{align*} \]

For $y=A\tan(B(x-C))+D$ or $y=A\cot(B(x-C))+D$:

\[ \begin{align*} \text{Period} &= \frac{\pi}{|B|} \\ \text{Midline} &= y=D \end{align*} \]

The vertical asymptotes are found by setting the argument equal to where the parent function is undefined.

Example: Analyze a tangent function

Find the stretching factor, period, phase shift, vertical shift, domain, and range of

$$ y=2\tan\left(\frac{x}{3}\right). $$

Show Solution

Match it to the standard form $y=A\tan(B(x-C))+D$.

\[ \begin{align*} A&=2 \\ B&=\frac{1}{3} \\ C&=0 \\ D&=0 \end{align*} \]

From there the features fall out.

\[ \begin{align*} \text{Stretching Factor} &= 2 \\ \text{Period} &= \frac{\pi}{\frac{1}{3}}=3\pi \\ \text{Phase Shift} &= 0 \\ \text{Vertical Shift} &= 0 \\ \text{Midline} &= y=0 \end{align*} \]

To locate one central branch, set the argument between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$:

$$ -\frac{\pi}{2}<\frac{x}{3}<\frac{\pi}{2} $$

Multiply through by 3:

$$ -\frac{3\pi}{2}<x<\frac{3\pi}{2} $$

So the asymptotes for the central branch are at $x=-\frac{3\pi}{2}$ and $x=\frac{3\pi}{2}$, and in general at $x=\frac{3\pi}{2}+3\pi k$ for any integer $k$.

\[ \begin{align*} \text{Domain} &= \left\{x \mid x \ne \frac{3\pi}{2}+3\pi k,\ k\in\mathbb{Z}\right\} \\ \text{Range} &= (-\infty,\infty) \end{align*} \]

Example: Analyze a cotangent function

Find the stretching factor, period, phase shift, vertical shift, domain, and range of

$$ y=3\cot\left(3x-\frac{3\pi}{2}\right)-2. $$

Show Solution

First rewrite the inside in phase-shift form.

\[ \begin{align*} 3x-\frac{3\pi}{2} &=3\left(x-\frac{\pi}{2}\right) \end{align*} \]

So the function is

$$ y=3\cot\left(3\left(x-\frac{\pi}{2}\right)\right)-2. $$

Now identify the constants.

\[ \begin{align*} A&=3 \\ B&=3 \\ C&=\frac{\pi}{2} \\ D&=-2 \end{align*} \]

Therefore,

\[ \begin{align*} \text{Stretching Factor} &= 3 \\ \text{Period} &= \frac{\pi}{3} \\ \text{Phase Shift} &= \frac{\pi}{2}\text{ right} \\ \text{Vertical Shift} &= 2\text{ down} \\ \text{Midline} &= y=-2 \end{align*} \]

Cotangent is undefined when its argument equals $\pi k$. Setting $3\left(x-\frac{\pi}{2}\right)=\pi k$ gives $x=\frac{\pi}{2}+\frac{\pi}{3}k$.

\[ \begin{align*} \text{Domain} &= \left\{x \mid x \ne \frac{\pi}{2}+\frac{\pi}{3}k,\ k\in\mathbb{Z}\right\} \\ \text{Range} &= (-\infty,\infty) \end{align*} \]

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Secant and Cosecant

Secant and cosecant piggyback on the cosine and sine graphs — their asymptotes fall exactly where the related function is zero. A good way to sketch them is to draw the guide sine or cosine lightly first, then build the branches around it.

Fact: Secant and Cosecant Features

For $y=A\sec(B(x-C))+D$ and $y=A\csc(B(x-C))+D$:

\[ \begin{align*} \text{Stretching Factor} &= |A| \\ \text{Period} &= \frac{2\pi}{|B|} \\ \text{Midline} &= y=D \end{align*} \]

The asymptotes occur where the related cosine or sine function equals zero.

Example: Analyze a secant function

Find the stretching factor, period, phase shift, vertical shift, domain, and range of

$$ y=2\sec\left(\frac{x}{2}-\pi\right)+1. $$

Show Solution

Rewrite the inside:

\[ \begin{align*} \frac{x}{2}-\pi &=\frac{1}{2}(x-2\pi) \end{align*} \]

So the function is

$$ y=2\sec\left(\frac{1}{2}(x-2\pi)\right)+1. $$

That gives us

\[ \begin{align*} A&=2 \\ B&=\frac{1}{2} \\ C&=2\pi \\ D&=1 \end{align*} \]

So the features are

\[ \begin{align*} \text{Stretching Factor} &= 2 \\ \text{Period} &= \frac{2\pi}{\frac{1}{2}}=4\pi \\ \text{Phase Shift} &= 2\pi\text{ right} \\ \text{Vertical Shift} &= 1\text{ up} \\ \text{Midline} &= y=1 \end{align*} \]

The asymptotes fall where $\cos\left(\frac{1}{2}(x-2\pi)\right)=0$, i.e., $\frac{1}{2}(x-2\pi)=\frac{\pi}{2}+\pi k$, giving $x=3\pi+2\pi k$.

\[ \begin{align*} \text{Domain} &= \left\{x \mid x \ne 3\pi+2\pi k,\ k\in\mathbb{Z}\right\} \\ \text{Range} &= (-\infty,-1]\cup[3,\infty) \end{align*} \]

Example: Analyze a cosecant function

Find the stretching factor, period, phase shift, vertical shift, domain, and range of

$$ y=-\csc\left(\frac{x}{3}\right)+4. $$

Show Solution

Compare it to $y=A\csc(B(x-C))+D$.

\[ \begin{align*} A&=-1 \\ B&=\frac{1}{3} \\ C&=0 \\ D&=4 \end{align*} \]

Now read off the features.

\[ \begin{align*} \text{Stretching Factor} &= 1 \\ \text{Period} &= \frac{2\pi}{\frac{1}{3}}=6\pi \\ \text{Phase Shift} &= 0 \\ \text{Vertical Shift} &= 4\text{ up} \\ \text{Midline} &= y=4 \end{align*} \]

Since $A<0$ there is also a reflection. The asymptotes occur where $\sin(x/3)=0$, i.e., $x=3\pi k$.

\[ \begin{align*} \text{Domain} &= \left\{x \mid x \ne 3\pi k,\ k\in\mathbb{Z}\right\} \\ \text{Range} &= (-\infty,3]\cup[5,\infty) \end{align*} \]

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