Graphing Secant and Cosecant

Learn to graph the secant and cosecant functions by understanding their relationship to cosine and sine.

Advanced25 min16 problems
Practice Problems

Definition

The secant and cosecant functions are the reciprocals of the cosine and sine functions:
sec⁡(x)=1cos⁡(x)\sec(x) = \frac{1}{\cos(x)}
csc⁡(x)=1sin⁡(x)\csc(x) = \frac{1}{\sin(x)}

Key Properties

**Secant Function y=sec⁡(x)y = \sec(x):**
  • Domain: All real numbers except x=π2+nπx = \frac{\pi}{2} + n\pi (where cos⁡(x)=0\cos(x) = 0)
  • Range: (−∞,−1]∪[1,∞)(-\infty, -1] \cup [1, \infty)
  • Period: 2π2\pi
  • Vertical asymptotes: At x=π2+nπx = \frac{\pi}{2} + n\pi
**Cosecant Function y=csc⁡(x)y = \csc(x):**
  • Domain: All real numbers except x=nπx = n\pi (where sin⁡(x)=0\sin(x) = 0)
  • Range: (−∞,−1]∪[1,∞)(-\infty, -1] \cup [1, \infty)
  • Period: 2π2\pi
  • Vertical asymptotes: At x=nπx = n\pi
Both functions have U-shaped curves that open upward (above y=1y = 1) or downward (below y=−1y = -1), never crossing the region between y=−1y = -1 and y=1y = 1.

Try it now

What is sec⁡(x)\sec(x) equal to?

Worked Examples

Sketch the graph of y=sec⁡(x)y = \sec(x) over the interval [−2π,2π][-2\pi, 2\pi].
1

First, sketch y=cos⁡(x)y = \cos(x)

Draw the cosine wave with maxima at x=0,±2πx = 0, \pm 2\pi and minima at x=±πx = \pm \pi → Reference curve drawn
2

Identify where cos⁡(x)=0\cos(x) = 0

This occurs at x=±π2,±3π2x = \pm\frac{\pi}{2}, \pm\frac{3\pi}{2} → Asymptote locations found
3

Draw vertical asymptotes

Draw dashed vertical lines at x=±π2,±3π2x = \pm\frac{\pi}{2}, \pm\frac{3\pi}{2} → 4 asymptotes drawn
4

Plot key points

sec⁡(0)=1\sec(0) = 1, sec⁡(π)=−1\sec(\pi) = -1, sec⁡(2π)=1\sec(2\pi) = 1 → Points at (0,1)(0, 1), (π,−1)(\pi, -1), (2π,1)(2\pi, 1)
5

Draw U-shaped curves

Between each pair of asymptotes, draw curves opening away from the x-axis → Complete secant graph

Common Mistakes

Placing asymptotes at the wrong locations

Why it's wrong: Students confuse where sine vs cosine equal zero. Secant has asymptotes where cos⁡=0\cos = 0; cosecant has asymptotes where sin⁡=0\sin = 0.

Correct: Remember: sec⁡=1cos⁡\sec = \frac{1}{\cos}, so asymptotes at cos⁡=0\cos = 0 (odd multiples of π2\frac{\pi}{2}). csc⁡=1sin⁡\csc = \frac{1}{\sin}, so asymptotes at sin⁡=0\sin = 0 (multiples of π\pi).

Drawing secant/cosecant curves crossing through y=0y = 0

Why it's wrong: Since ∣sec⁡(x)∣≥1|\sec(x)| \geq 1 and ∣csc⁡(x)∣≥1|\csc(x)| \geq 1, these functions never equal zero.

Correct: The curves only exist for y≤−1y \leq -1 or y≥1y \geq 1. There is always a gap between −1-1 and 11.

Forgetting that the period is 2π2\pi, not π\pi

Why it's wrong: Students may confuse with tangent/cotangent which have period π\pi.

Correct: Secant and cosecant have period 2π2\pi, same as their reciprocals (cosine and sine).

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How to Graph Cosecant, Secant, and Cotangent Functions | Examples

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Interactive Visual

Function graph

-2π-π0π2π-4-2024xyx = -3π/2x = -π/2x = π/2x = 3π/2

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Asymptote

Function graph

-2π-π0π2π-4-2024xyx = -2πx = -πx = 0x = πx = 2π

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Asymptote

Interactive Sandbox

Interactive Grapher

-3π-2π-π0π2π3π-4-2024xy

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Practice Problems

16 problems
Problem 1 of 16
Easy
What is sec⁡(x)\sec(x) equal to?

Why It Matters

Secant and cosecant functions appear in many advanced applications:
  • Maps: On a Mercator world map, distances at latitude φ\varphi are stretched by the factor sec⁡(φ)\sec(\varphi), which is why Greenland looks so big and the poles (the asymptote at 90°90°) never fit on the map
  • Engineering: Analyzing electrical circuits with alternating current
  • Architecture: Calculating structural loads and tension in cables
  • Navigation: Computing distances and angles in spherical trigonometry
  • Calculus: These functions are essential for integration techniques and solving differential equations
Understanding their graphs helps visualize behavior at critical points and prepares you for advanced mathematics.

Real World Applications

Tracking an Airplane

Radar and air traffic control turn an elevation angle into a straight-line distance with cosecant: a plane at height $h$ seen at elevation angle $\theta$ is $h \cdot \csc(\theta)$ away.

Example:

A plane flying 3 km high is seen at an elevation angle of 30°30°. Its straight-line distance is 3csc⁡(30°)=3⋅2=63\csc(30°) = 3 \cdot 2 = 6 km.

1Try It Yourself

A plane flies level at a height of 3 km and passes right over you. Its distance from you is d(θ)=3csc⁡(θ)d(\theta) = 3\csc(\theta) km, where θ\theta is the elevation angle, 0<θ<π0 < \theta < \pi.

What is the smallest distance between you and the plane, in km?

Step 1: Write the mathematical expression

Write the smallest value of 3csc⁡(θ)3\csc(\theta):

Structural Engineering

Engineers use secant to find the pulling force in a slanted cable that holds up a load.

Example:

If a cable makes angle θ\theta with the vertical and its pull must carry a weight WW (a horizontal bar takes the sideways push), the force in the cable is T=W⋅sec⁡(θ)T = W \cdot \sec(\theta).

2Try It Yourself

A shop sign weighing 500 N hangs from a wall bar and a cable at 60°60° from the vertical. The force in the cable is T=500⋅sec⁡(60°)T = 500 \cdot \sec(60°) newtons.

What is sec⁡(60°)\sec(60°)?

Step 1: Write the mathematical expression

Calculate sec⁡(60°)=1cos⁡(60°)\sec(60°) = \frac{1}{\cos(60°)}:

Key Takeaways

  • 1Secant is the reciprocal of cosine: sec⁡(x)=1cos⁡(x)\sec(x) = \frac{1}{\cos(x)}
  • 2Cosecant is the reciprocal of sine: csc⁡(x)=1sin⁡(x)\csc(x) = \frac{1}{\sin(x)}
  • 3Both have vertical asymptotes where their reciprocal functions equal zero
  • 4Range is (−∞,−1]∪[1,∞)(-\infty, -1] \cup [1, \infty): they never take values between −1-1 and 11
  • 5Period of both functions is 2π2\pi
  • 6Graph by first sketching the reciprocal function (cos or sin), then drawing U-shaped curves

Frequently Asked Questions

They are reciprocals because sec⁡(x)=1cos⁡(x)\sec(x) = \frac{1}{\cos(x)} and csc⁡(x)=1sin⁡(x)\csc(x) = \frac{1}{\sin(x)}. Wherever sine or cosine equals some value vv, the corresponding reciprocal function equals 1v\frac{1}{v}.
They are reciprocals because sec⁡(x)=1cos⁡(x)\sec(x) = \frac{1}{\cos(x)} and csc⁡(x)=1sin⁡(x)\csc(x) = \frac{1}{\sin(x)}. Wherever sine or cosine equals some value vv, the corresponding reciprocal function equals 1v\frac{1}{v}.
Each pair has exactly one "co": secant goes with cosine, and cosecant goes with sine. So secant has asymptotes where cos⁡=0\cos = 0, and cosecant has asymptotes where sin⁡=0\sin = 0.
No. Since ∣cos⁡(x)∣≤1|\cos(x)| \leq 1 and ∣sin⁡(x)∣≤1|\sin(x)| \leq 1, their reciprocals always satisfy ∣sec⁡(x)∣≥1|\sec(x)| \geq 1 and ∣csc⁡(x)∣≥1|\csc(x)| \geq 1. They never equal zero.

Glossary

Secant function
The reciprocal of cosine: sec⁡(x)=1cos⁡(x)\sec(x) = \frac{1}{\cos(x)}
Cosecant function
The reciprocal of sine: csc⁡(x)=1sin⁡(x)\csc(x) = \frac{1}{\sin(x)}
Vertical asymptote
A vertical line that the graph approaches but never touches, occurring where the function is undefined
Period
The horizontal length after which a function repeats; 2π2\pi for secant and cosecant
Reciprocal function
A function that equals 1f(x)\frac{1}{f(x)} for some function f(x)f(x)
Today's challengeOne problem, three tries

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