Unit 5 · Topic 3.8
The Tangent Function
Build tangent from slope and the unit circle, then analyze its repeating branches, asymptotes, and transformations.
Learning Goals
- Interpret tangent as the slope of a terminal ray.
- Connect \(\tan\theta\) to \(\sin\theta/\cos\theta\).
- Identify the domain, range, period, zeros, asymptotes, symmetry, and concavity of tangent.
- Analyze transformations in \(y=A\tan(B(x-C))+D\).
- Construct a tangent equation from graph features and reference points.
1. Tangent Is the Slope of a Ray
Let the terminal ray of an angle \(\theta\) pass through a nonzero point \((x,y)\). Its slope is
Every point on the same ray gives the same ratio because scaling both coordinates does not change the slope. A vertical terminal ray has no finite slope, so tangent is undefined there.
2. Connect Tangent to Sine and Cosine
On a circle of radius \(r\), the terminal point is \((r\cos\theta,r\sin\theta)\). Therefore,
This quotient explains both the signs of tangent and every input where tangent is undefined.
3. Exact Values and Signs
| \(\theta\) | 0 | \(\pi/6\) | \(\pi/4\) | \(\pi/3\) | \(\pi/2\) |
|---|---|---|---|---|---|
| \(\tan\theta\) | 0 | \(\sqrt3/3\) | 1 | \(\sqrt3\) | Undefined |
Tangent is positive in Quadrants I and III, where sine and cosine have the same sign. It is negative in Quadrants II and IV, where their signs differ.
4. Why the Period Is \(\pi\)
Rotating a terminal ray by \(\pi\) reverses both coordinates but preserves the slope:
Thus the fundamental period is \(\pi\), one half-revolution. It is not the \(2\pi\) period of sine and cosine.
5. Graph the Parent Function
Each branch lies between consecutive vertical asymptotes, crosses the x-axis midway between them, and repeats every \(\pi\) units.
6. Parent-Function Characteristics
| Feature | \(y=\tan x\) |
|---|---|
| Domain | \(x\ne\pi/2+k\pi,\ k\in\mathbb Z\) |
| Range | \((-\infty,\infty)\) |
| Period | \(\pi\) |
| Zeros | \(x=k\pi\) |
| Vertical asymptotes | \(x=\pi/2+k\pi\) |
| Amplitude | None |
Tangent has no maximum, minimum, or amplitude because each branch takes every real output value.
7. Increasing Behavior and Concavity
On every interval between consecutive asymptotes, tangent increases from outputs approaching \(-\infty\) to outputs approaching \(+\infty\).
Each parent branch is concave down before its center \(x=k\pi\), changes concavity there, and is concave up after the center. The point \((k\pi,0)\) is a point of inflection.
8. Tangent Is an Odd Function
The parent graph has rotational symmetry about the origin. More generally, a translated tangent graph has 180-degree rotational symmetry about the center of each branch.
9. Use Factored Transformation Form
| Parameter | Effect |
|---|---|
| \(A\) | Vertical dilation; reflection when negative |
| \(B\) | Horizontal dilation and period |
| \(C\) | Horizontal shift and branch center |
| \(D\) | Vertical shift and center-line output |
Unlike a sinusoidal model, \(|A|\) is a vertical scale factor rather than an amplitude.
10. Period, Centers, and Asymptotes
Branch centers repeat at \(x=C+kP\). The two neighboring asymptotes are one half-period from a center:
Factoring the input first prevents an incorrect phase shift.
11. Three Reference Points Determine a Branch
For \(B>0\), use the center and the quarter-period inputs:
| Input | Inside angle | Output |
|---|---|---|
| \(C-P/4\) | \(-\pi/4\) | \(D-A\) |
| \(C\) | 0 | \(D\) |
| \(C+P/4\) | \(\pi/4\) | \(D+A\) |
If \(A<0\), the outputs reverse and the branch decreases. Draw the asymptotes before connecting the reference points.
12. Worked Transformation
Analyze
- Period: \(P=\pi/3\)
- Branch center: \((\pi/6,1)\)
- Nearest asymptotes: \(x=0\) and \(x=\pi/3\)
- Reference points: \((\pi/12,3)\), \((\pi/6,1)\), and \((\pi/4,-1)\)
- Range: all real numbers
The negative value of \(A\) reflects the parent graph vertically, so the branch decreases between its asymptotes.
13. Construct an Equation from a Graph
Suppose consecutive asymptotes are \(x=-2\) and \(x=4\), the center is \((1,-1)\), and the graph also passes through \((2.5,3)\).
Since \(2.5=C+P/4\), the tangent input is \(\pi/4\), so
14. Read a Tangent Graph Efficiently
- Measure the distance between consecutive asymptotes or centers to find \(P\).
- Calculate \(|B|=\pi/P\).
- Locate the midpoint between asymptotes to find \(C\).
- Read the center output to find \(D\).
- Use a quarter-period point to determine \(A\) and direction.
- Verify the next asymptote and center by periodicity.
15. Common Errors
- Assigning tangent a period of \(2\pi\) instead of \(\pi\).
- Calling \(|A|\) an amplitude even though tangent is unbounded.
- Including asymptote inputs in the domain.
- Drawing separate branches across a vertical asymptote as one connected curve.
- Reading \(C\) before factoring \(B(x-C)\).
- Placing asymptotes a full period from the center instead of a half-period.
- Assuming every transformed tangent function is odd about the origin.
Key Takeaways
- Analyze tangent through its period, zeros, asymptotes, signs, and transformations.
- Core relationship: \(\tan x=\frac{\sin x}{\cos x},\quad P=\pi\)
- Error check: Do not give tangent the \(2\pi\) period of sine and cosine.
- Explain why \(\tan\theta\) is undefined when \(\cos\theta=0\).
- State the domain, range, period, zeros, and asymptotes of \(y=\tan x\).
- Analyze \(y=3\tan(2(x+\pi/4))-2\).
- List the center, adjacent asymptotes, and quarter-period points of \(y=-\tan(\pi(x-1))+4\).
- Construct a tangent equation with asymptotes \(x=0\) and \(x=8\), center output 2, and point \((6,5)\).