The Tangent Function (College Board AP® Precalculus): Study Guide

The tangent function & the unit circle

How is the tangent function defined using the unit circle?

• Given an angle $\theta$ in standard position and a unit circle centered at the origin

  • the terminal ray intersects the circle at a point $P$

• The value of the tangent function, $f\left(\theta \right)=\tan \theta$

  • is given by the slope of the terminal ray

• This connects directly to the unit circle coordinates of point $P$

  • $P$ has coordinates $(\cos \theta ,\sin \theta )$

  • So the slope of a line through the origin $\left(0,0\right)$ and the point $P$ is

    • $\text{slope}=\frac{\sin \theta -0}{\cos \theta -0}=\frac{\sin \theta }{\cos \theta }$

• Therefore

  • $\tan \theta =\frac{\sin \theta }{\cos \theta },\text{where }\cos \theta \ne 0$

• This slope interpretation means

  • When the terminal ray is steep and rising

    • $\tan \theta$ is a large positive number

  • When the terminal ray is horizontal (along the $x$-axis)

    • $\tan \theta =0$

  • When the terminal ray is steep and falling

    • $\tan \theta$ is a large negative number

  • When the terminal ray is vertical

    • the slope is undefined

    • so $\tan \theta$ is undefined

What are the exact values of tangent for key angles?

• Since $\tan \theta =\frac{\sin \theta }{\cos \theta }$, the exact values can be calculated from the sine and cosine values

• Because tangent has a period of $\pi$ , the values repeat every $\pi$ radians

  • E.g. $\tan \frac{7\pi }{6}=\tan \frac{\pi }{6}=\frac{\sqrt{3}}{3}$

Key characteristics of the tangent function

What is the period of the tangent function?

• The slope values of the terminal ray repeat every half revolution of the unit circle

  • For example, the terminal rays at $\theta$ and $\theta +\pi$ point in opposite directions

    • but they lie on the same line

    • and therefore have the same slope

• Therefore the tangent function has a period of $\pi$:

$\tan (\theta +\pi )=\tan \theta$

• This is different from sine and cosine

  • which both have a period of $2\pi$

Where does the tangent function have asymptotes?

• The tangent function is undefined when $\mathbf{cos}\mathit{\theta }\mathbf{=}\mathbf{0}$, which occurs at

$\theta =\frac{\pi }{2}+k\pi ,\mathrm{for}\mathrm{integer}\mathrm{values}\mathrm{of}k$

• At these input values, the terminal ray is vertical (pointing straight up or straight down)

  • so its slope does not exist

• The graph of the tangent function has vertical asymptotes at each of these values

  • E.g. as $\theta$ approaches $\frac{\pi }{2}$ from the left, $\tan \theta$ increases without bound

  • and as $\theta$ approaches $\frac{\pi }{2}$ from the right, $\tan \theta$ decreases without bound

• The tangent function therefore demonstrates periodic asymptotic behavior

  • the pattern of asymptotes repeats with the same period as the function

What is the behavior of the tangent function between consecutive asymptotes?

• Between any two consecutive asymptotes

  • the tangent function is always increasing

• The graph changes from concave down to concave up between consecutive asymptotes

  • Specifically, the graph is

    • concave down on the first half of the interval

    • and concave up on the second half

  • The changeover occurs at the point of inflection

    • which is where $\tan \theta =0$

    • i.e. at integer multiples of $\pi$

• Between consecutive asymptotes, the tangent function takes all real number output values, from $-\infty$ to $+\infty$

Transformations of the tangent function

How do additive transformations affect the tangent function?

• A vertical translation of the tangent function is given by $g\left(\theta \right)=\tan \theta +d$

  • This shifts the graph vertically by $d$ units

  • The line containing the points of inflection is also shifted up by $d$ units

    • from $y=0$ to $y=d$

  • The asymptotes are not affected by a vertical translation

• A horizontal translation (phase shift) is given by $g\left(\theta \right)=\tan (\theta +c)$

  • This shifts the graph horizontally by $-c$ units

    • to the left if $c>0$, to the right if $c<0$

  • The asymptotes shift by $-c$ units as well

How do multiplicative transformations affect the tangent function?

• A vertical dilation is given by $g\left(\theta \right)=a\tan \theta$

  • This stretches the graph vertically by a factor of $\left|a\right|$

  • If $\mathit{a}\mathbf{<}\mathbf{0}$, the graph is also reflected over the $x$-axis

    • this causes the function to be decreasing (instead of increasing) between consecutive asymptotes

    • and reverses the concavity pattern

  • The asymptotes and period are not affected

• A horizontal dilation is given by $g\left(\theta \right)=\tan \left(b\theta \right)$

  • This changes the period of the function by a factor of $\frac{1}{\left|b\right|}$

    • The new period is $\frac{\pi }{\left|b\right|}$

  • If $\mathit{b}\mathbf{<}\mathbf{0}$, the graph is also reflected over the $y$-axis

    • this causes the function to be decreasing (instead of increasing) between consecutive asymptotes

    • and reverses the concavity pattern

  • The asymptotes are compressed or stretched accordingly

How are all transformations combined?

• The general form combining all transformations is $f\left(\theta \right)=a\tan \left(b\right(\theta +c\left)\right)+d$

• The parameters have the following effects

  • $\left|a\right|$: vertical dilation factor (with reflection over the $x$-axis if $a<0$)

  • $\frac{\pi }{\left|b\right|}$: the period of the function

  • $-c$: the phase shift (horizontal translation)

  • $d$: the vertical shift (the new midline of the points of inflection is $y=d$)

• For example, $g\left(\theta \right)=3\tan \left(2\right(\theta -\frac{\pi }{4}\left)\right)+1$ has:

  • Vertical dilation factor $3$

  • Period $=\frac{\pi }{2}$

  • Phase shift of $\frac{\pi }{4}$ to the right (since $c=-\frac{\pi }{4}$)

  • Vertical shift of $1$ unit up