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Reciprocal Trigonometric Functions (College Board AP® Precalculus): Study Guide

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 15 April 2026

Secant, cosecant & cotangent

What is the secant function?

The secant function is the reciprocal of the cosine function
- $\sec θ = \frac{1}{\cos θ}, where \cos θ \neq 0$

The secant function is undefined wherever $\cos θ = 0$
- i.e. at $θ = \frac{π}{2} + k π$ for integer values of $k$

What is the cosecant function?

The cosecant function is the reciprocal of the sine function
- $\csc θ = \frac{1}{\sin θ}, where \sin θ \neq 0$

The cosecant function is undefined wherever $\sin θ = 0$
- i.e. at $θ = k π$ for integer values of $k$

What are the key characteristics of the secant and cosecant graphs?

The graphs of secant and cosecant have vertical asymptotes at the values where cosine and sine are zero, respectively
- $\sec θ$ has vertical asymptotes at $θ = \frac{π}{2} + k π$
  - the same values where $\cos θ = 0$
- $\csc θ$ has vertical asymptotes at $θ = k π$
  - the same values where $\sin θ = 0$
The range of both secant and cosecant is $(- \infty, - 1] \cup [1, \infty)$
- Output values are always $\leq - 1$ or $\geq 1$
  - the functions will never output values between $- 1$ and $1$
- This makes sense because they are reciprocals of functions whose outputs lie between $- 1$ and $1$
  - Taking the reciprocal of a number with absolute value at most $1$ gives a number with absolute value at least $1$
Both functions are periodic
- secant has period $2 π$ (same as cosine)
- and cosecant has period $2 π$ (same as sine)

Graph of y = sec(θ) showing periodic curves across angles from -450° to 450°, with asymptotes at odd multiples of 90°. — **Graph of y=secθ**

Graph of y = csc(θ) showing periodic curves between asymptotes at -360°, -180°, 0°, 180°, 360°, with local extrema at ±1 along the y-axis. — **Graph of y=cscθ**

What is the cotangent function?

The cotangent function is the reciprocal of the tangent function
- $\cot θ = \frac{1}{\tan θ}, where \tan θ \neq 0$

Equivalently, cotangent can be written as the ratio of cosine to sine
- $\cot θ = \frac{\cos θ}{\sin θ}, where \sin θ \neq 0$

The cotangent function is undefined wherever $\sin θ = 0$
- i.e. at $θ = k π$ for integer values of $k$

What are the key characteristics of the cotangent graph?

The graph of cotangent has vertical asymptotes at values where $\tan θ = 0$
- i.e. at $θ = k π$ for integer values of $k$
- Note that these are the same values where $\sin θ = 0$
  - which is consistent with the formula $\cot θ = \frac{\cos θ}{\sin θ}$
Between consecutive asymptotes, the cotangent function is always decreasing
- This is the opposite of tangent, which is always increasing between its consecutive asymptotes
The cotangent function has a period of $π$ (the same as tangent)
The range of cotangent is all real numbers (the same as tangent)

Graph of the cotangent function y=cotθ with vertical asymptotes at -360°, -180°, 0°, 180°, 360°. The curve crosses the x-axis at ±90°. — **Graph of y=cotθ**

How can the reciprocal functions be evaluated at key angles?

To evaluate secant, cosecant, or cotangent at a specific angle
- first find the value of the corresponding base function (cosine, sine, or tangent)
- then take the reciprocal
For example:
- $\sec \frac{π}{3} = \frac{1}{\cos \frac{π}{3}} = \frac{1}{\frac{1}{2}} = 2$
- $\csc \frac{π}{4} = \frac{1}{\sin \frac{π}{4}} = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}$
- $\cot \frac{π}{6} = \frac{1}{\tan \frac{π}{6}} = \frac{1}{\frac{\sqrt{3}}{3}} = \frac{3}{\sqrt{3}} = \sqrt{3}$
If the base function equals zero at that angle, the reciprocal function is undefined
- E.g. $\csc π$ is undefined because $\sin π = 0$

Examiner Tips and Tricks

A quick way to remember where each reciprocal function has asymptotes is to think about where its "partner" base function equals zero.

Secant's asymptotes are where cosine is zero
cosecant's asymptotes are where sine is zero
and cotangent's asymptotes are also where sine is zero (because $\cot θ = \frac{\cos θ}{\sin θ}$ )

Knowing the zeros of sine and cosine from the unit circle makes it straightforward to identify these asymptote locations.

Worked Example

Find the exact value of each of the following expressions, or state that it is undefined.

(a) $\sec \frac{2 π}{3}$

Answer:

The secant is the reciprocal of cosine

$\sec \frac{2 π}{3} = \frac{1}{\cos \frac{2 π}{3}} = \frac{1}{(- \frac{1}{2})} = - 2$

$\sec \frac{2 π}{3} = - 2$

(b) $\csc \frac{π}{3}$

Answer:

The cosecant is the reciprocal of sine:

$\csc \frac{π}{3} = \frac{1}{\sin \frac{π}{3}} = \frac{1}{(\frac{\sqrt{3}}{2})} = \frac{2}{\sqrt{3}} = \frac{2 \sqrt{3}}{3}$

$\csc \frac{π}{3} = \frac{2 \sqrt{3}}{3}$

Answer:

The cotangent can be found as the reciprocal of tangent, or as $\frac{\cos θ}{\sin θ}$

$\cot \frac{3 π}{4} = \frac{\cos \frac{3 π}{4}}{\sin \frac{3 π}{4}} = \frac{- \frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = - 1$

$\cot \frac{3 π}{4} = - 1$

(d) $\csc π$

Answer:

The cosecant is the reciprocal of sine

$\csc π = \frac{1}{\sin π} = \frac{1}{0}$

Since $\sin π = 0$ , the expression $\csc π$ is undefined

The graph of cosecant has a vertical asymptote at $θ = π$

$\csc π is undefined$

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Reciprocal Trigonometric Functions (College Board AP® Precalculus): Study Guide

Secant, cosecant & cotangent

What is the secant function?

What is the cosecant function?

What are the key characteristics of the secant and cosecant graphs?

What is the cotangent function?

What are the key characteristics of the cotangent graph?

How can the reciprocal functions be evaluated at key angles?

Examiner Tips and Tricks

Worked Example

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Join the 100,000+ Students that ❤️ Save My Exams

Author: Roger B

Reviewer: Mark Curtis