Graphing the Tangent Function

Learn to graph y = tan(x), identify asymptotes, period, and key features of the tangent curve.

Advanced25 minLesson
Printable worksheet

Definition

The tangent function is defined as the ratio of sine to cosine:
tan⁡(x)=sin⁡(x)cos⁡(x)\tan(x) = \frac{\sin(x)}{\cos(x)}
Unlike sine and cosine, the tangent function has vertical asymptotes where cos⁡(x)=0\cos(x) = 0.
**Key Properties of y=tan⁡(x)y = \tan(x):**
  • Period: π\pi (not 2π2\pi like sine and cosine)
  • Domain: All real numbers except x=π2+nπx = \frac{\pi}{2} + n\pi where nn is any integer
  • Range: All real numbers (−∞,∞)(-\infty, \infty)
  • Asymptotes: Vertical lines at x=π2,3π2,−π2,...x = \frac{\pi}{2}, \frac{3\pi}{2}, -\frac{\pi}{2}, ...
  • x-intercepts: At x=0,±π,±2π,...x = 0, \pm\pi, \pm 2\pi, ...
  • The function passes through the origin (0,0)(0, 0)

Try it now

What is the period of y=tan⁡(x)y = \tan(x)?

Worked Examples

Sketch one period of y=tan⁡(x)y = \tan(x) and identify the asymptotes, intercepts, and behavior.
1

Find the asymptotes

cos⁡(x)=0\cos(x) = 0 when x=−π2x = -\frac{\pi}{2} and x=π2x = \frac{\pi}{2} → Vertical asymptotes at x=±π2x = \pm\frac{\pi}{2}
2

Find the x-intercept

tan⁡(x)=0\tan(x) = 0 when sin⁡(x)=0\sin(x) = 0, so x=0x = 0 → x-intercept at (0,0)(0, 0)
3

Find key points

tan⁡(−π4)=−1\tan\left(-\frac{\pi}{4}\right) = -1 and tan⁡(π4)=1\tan\left(\frac{\pi}{4}\right) = 1 → Points: (−π4,−1)\left(-\frac{\pi}{4}, -1\right) and (π4,1)\left(\frac{\pi}{4}, 1\right)
4

Describe the behavior

As x→−π2+x \to -\frac{\pi}{2}^+, tan⁡(x)→−∞\tan(x) \to -\infty; as x→π2−x \to \frac{\pi}{2}^-, tan⁡(x)→+∞\tan(x) \to +\infty → Curve rises from −∞-\infty to +∞+\infty

Common Mistakes

Thinking the period of tangent is 2π2\pi like sine and cosine

Why it's wrong: The tangent function repeats every π\pi radians because tan⁡(x+π)=tan⁡(x)\tan(x + \pi) = \tan(x) for all xx in the domain.

Correct: The period of y=tan⁡(x)y = \tan(x) is π\pi. For y=tan⁡(bx)y = \tan(bx), the period is π∣b∣\frac{\pi}{|b|}.

Drawing the curve crossing through the asymptotes

Why it's wrong: Asymptotes represent values where the function is undefined. The curve approaches but never touches or crosses them.

Correct: Draw vertical dashed lines for asymptotes. The curve approaches ±∞\pm\infty as it nears each asymptote.

Forgetting that tangent has no maximum or minimum value

Why it's wrong: Unlike sine and cosine which are bounded between −1-1 and 11, tangent's range is all real numbers.

Correct: The range of y=tan⁡(x)y = \tan(x) is (−∞,∞)(-\infty, \infty). There is no amplitude for tangent functions.

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

How to Graph a General Tangent Function | Step-By-Step Example

by Understand The Math

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Interactive Visual

Function graph

-2π-π0π2π-4-2024xy(-6.28, 0)(-3.14, 0)(0, 0)(3.14, 0)(6.28, 0)x = -3π/2x = -π/2x = π/2x = 3π/2

Tap or hover over the graph to read off values.

  • Period = π
y = A·tan(Bx)Asymptote
  • Zero (x-intercept): (-6.28, 0)
  • Zero (x-intercept): (-3.14, 0)
  • Zero (x-intercept): (0, 0)
  • Zero (x-intercept): (3.14, 0)
  • Zero (x-intercept): (6.28, 0)

Unit Circle

Degrees

0°

Radians

0

sin(θ)

0

cos(θ)

1

tan(θ)

0

Coordinates (cos, sin)

(1, 0)

Click and drag to rotate the angle around the unit circle.

Interactive Sandbox

Interactive Grapher

-3π-2π-π0π2π3π-4-2024xy

Tap or hover over the graph to read off values.

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Practice Problems

15 problems
Problem 1 of 15
Easy
What is the period of y=tan⁡(x)y = \tan(x)?

Why It Matters

The tangent function appears throughout mathematics and real-world applications:
  • Architecture: Calculating roof pitch and ramp angles
  • Physics: Analyzing projectile motion and pendulum swings
  • Navigation: Determining bearings and flight paths
  • Engineering: Designing roads, bridges, and structures with specific inclines
Understanding the tangent graph helps you visualize how the ratio of vertical to horizontal change behaves as angles vary.

Real World Applications

Road Engineering

Engineers use the tangent function when designing roads with specific gradients and banking angles.

Example:

A road rises 5 meters over a horizontal distance of 100 meters. The grade angle θ\theta satisfies tan⁡(θ)=5100=0.05\tan(\theta) = \frac{5}{100} = 0.05, giving θ≈2.86°\theta \approx 2.86°.

1Try It Yourself

A wheelchair ramp must have a maximum slope ratio of 1:12 (rise to run).

What is the maximum angle of the ramp?

Step 1: Write the mathematical expression

Calculate arctan⁡(112)\arctan\left(\frac{1}{12}\right):

Aviation Navigation

Pilots use tangent ratios when calculating descent angles and approach paths.

Example:

A plane descends from 10,000 feet while traveling 30 miles horizontally. The descent angle θ\theta has tan⁡(θ)=1000030×5280\tan(\theta) = \frac{10000}{30 \times 5280}.

2Try It Yourself

A standard 3-degree glide slope is used for landing approaches.

For every mile of horizontal distance, how many feet does the plane descend?

Step 1: Write the mathematical expression

Calculate: 5280×tan⁡(3°)5280 \times \tan(3°)

Key Takeaways

  • 1The tangent function is defined as tan⁡(x)=sin⁡(x)cos⁡(x)\tan(x) = \frac{\sin(x)}{\cos(x)}
  • 2The period of y=tan⁡(x)y = \tan(x) is π\pi (not 2π2\pi)
  • 3Vertical asymptotes occur at x=π2+nπx = \frac{\pi}{2} + n\pi where cosine equals zero
  • 4The range is all real numbers (−∞,∞)(-\infty, \infty) with no maximum or minimum
  • 5For y=tan⁡(bx)y = \tan(bx), the period is π∣b∣\frac{\pi}{|b|}
  • 6The graph passes through the origin and has x-intercepts at multiples of π\pi

Frequently Asked Questions

Because tan⁡(x)=sin⁡(x)cos⁡(x)\tan(x) = \frac{\sin(x)}{\cos(x)}, the function is undefined wherever cos⁡(x)=0\cos(x) = 0. Division by zero creates vertical asymptotes. Sine and cosine are defined for all real numbers with no division involved.
Because tan⁡(x)=sin⁡(x)cos⁡(x)\tan(x) = \frac{\sin(x)}{\cos(x)}, the function is undefined wherever cos⁡(x)=0\cos(x) = 0. Division by zero creates vertical asymptotes. Sine and cosine are defined for all real numbers with no division involved.
Asymptotes occur where cosine equals zero: at odd multiples of π2\frac{\pi}{2}. That's ±π2,±3π2,±5π2\pm\frac{\pi}{2}, \pm\frac{3\pi}{2}, \pm\frac{5\pi}{2}, etc. Think: halfway between each zero of cosine.
The tangent function repeats after π\pi radians because both sine and cosine change sign together after π\pi, so their ratio stays the same: −sin⁡(x)−cos⁡(x)=sin⁡(x)cos⁡(x)\frac{-\sin(x)}{-\cos(x)} = \frac{\sin(x)}{\cos(x)}.

Glossary

Asymptote
A line that a curve approaches but never touches. For tangent, these are vertical lines where the function is undefined.
Period
The horizontal distance before a function repeats. For y=tan⁡(x)y = \tan(x), the period is π\pi.
Tangent function
A trigonometric function defined as tan⁡(x)=sin⁡(x)cos⁡(x)\tan(x) = \frac{\sin(x)}{\cos(x)}, representing the ratio of opposite to adjacent sides in a right triangle.
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