Topic 3.8 Notes – The Tangent Function

What Tangent Is

For an angle $\theta$ in standard position, the terminal ray hits the unit circle at
$P=(\cos\theta,\sin\theta)$.

That point gives tangent a clean meaning. The slope from the origin to $P$ is

$$\tan\theta=\frac{\sin\theta}{\cos\theta}, \qquad \cos\theta\ne 0$$

So tangent is the slope of the terminal ray.

Tangent on the unit circle

In the diagram, the green right triangle shows why $\tan\theta=\dfrac{\sin\theta}{\cos\theta}$. The red segment is another common unit-circle picture for tangent.

A few things come straight from that definition:

• If $\cos\theta=0$, the run is $0$, so the ray is vertical and tangent is undefined.
• The sign comes from the signs of sine and cosine:
  • Quadrant I: $+\!/\!+=+$
  • Quadrant II: $+\!/\!-= -$
  • Quadrant III: $-\!/\!-=+$
  • Quadrant IV: $-\!/\!+=-$
• The right-triangle memory aid opposite over adjacent still works, but the unit-circle definition works in all four quadrants.

The Parent Tangent Graph

The parent function is $y=\tan\theta$.

Three anchor values are worth knowing cold:

• $\tan\left(-\frac{\pi}{4}\right)=-1$
• $\tan(0)=0$
• $\tan\left(\frac{\pi}{4}\right)=1$

One basic branch lives on $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. As $\theta$ moves left to right there, the graph rises from very negative to very positive. The full graph just repeats that pattern every $\pi$, and these three anchor points show up in the center branch.

Key facts that get tested a lot:

• Vertical asymptotes at $\theta=\frac{\pi}{2}+k\pi$
• Period is $\pi$, not $2\pi$
• Domain excludes $\theta=\frac{\pi}{2}+k\pi$
• Range is all real numbers
• Zeros at $\theta=k\pi$, so intercepts are $(k\pi,0)$
• Odd symmetry: $\tan(-\theta)=-\tan\theta$
• Increasing on each interval between consecutive asymptotes
• Concavity
  • down on $\left(-\frac{\pi}{2}+k\pi,k\pi\right)$
  • up on $\left(k\pi,\frac{\pi}{2}+k\pi\right)$
• Inflection points at $(k\pi,0)$
• No amplitude, no maximum, no minimum

Transformations of Tangent

The general form is

$$y=a\tan(b(\theta+c))+d$$

Each parameter changes a different feature:

• $a$ changes vertical stretch/compression by factor $|a|$
  • if $a<0$, reflect across the $x$-axis
  • $|a|$ is not amplitude
• $b$ changes horizontal stretch/compression by factor $\frac{1}{|b|}$
  • new period is $\frac{\pi}{|b|}$
  • if $b<0$, reflect across the $y$-axis
• $c$ gives phase shift of $-c$
  • positive $c$ means left
  • negative $c$ means right
• $d$ shifts the graph up or down
  • the inflection-point line becomes $y=d$

That line $y=d$ matters, but it is not an asymptote and it is not a sine/cosine midline.

Graphing and Reading $y=a\tan(b(\theta+c))+d$

Use the inner input $u=b(\theta+c)$. Parent tangent is built around these key inputs:

1. $u=-\frac{\pi}{2}$ and $u=\frac{\pi}{2}$ give vertical asymptotes
2. $u=0$ gives the inflection point
3. $u=\pm\frac{\pi}{4}$ give anchor points with outputs $d-a$ and $d+a$

Then solve each equation for $\theta$ to place the features.

Important formulas:

• Asymptotes from $b(\theta+c)=\frac{\pi}{2}+k\pi$
• Inflection points at $\left(-c+\frac{k\pi}{b},\,d\right)$

Branch behavior:

• If $ab>0$, each branch is increasing
• If $ab<0$, each branch is decreasing
• If $b<0$, the parent key-input order reverses left to right

Common Exam Mistakes

• Using $2\pi$ as the period instead of $\pi$
• Saying tangent has amplitude or a midline
• Forgetting tangent is undefined where $\cos\theta=0$
• Treating an asymptote as part of the graph or saying tangent “equals infinity”
• Reading phase shift from an unfactored input like $\tan(2\theta-\pi/3)$ instead of rewriting it as $\tan\left(2\left(\theta-\pi/6\right)\right)$
• Saying the graph decreases across an asymptote instead of analyzing one branch at a time