# The secant, cosecant, and cotangent functions

> AP Precalculus · AP Precalculus 2024+ Framework
> Source: https://www.owlsprep.com/study/ap-precalculus-u3-the-secant-cosecant-and-cotangent/

This module covers definitions, unit circle evaluations, domain/range properties, key features, and graphing rules for the three reciprocal trigonometric functions: secant, cosecant, and cotangent.

**Prerequisites:** [Sine, cosine, and tangent definitions and unit circle values](https://www.owlsprep.com/study/ap-precalculus-u3-sine-cosine-tangent-functions/); [Vertical asymptote rules for rational functions](https://www.owlsprep.com/study/ap-precalculus-u1-rational-functions-key-features/)

## Learning objectives

- Define secant, cosecant, and cotangent as reciprocal trigonometric functions
- Derive domain, range, and vertical asymptote properties for each reciprocal function
- Evaluate reciprocal trigonometric values for standard unit circle angles
- Sketch transformed graphs of secant, cosecant, and cotangent for AP exam tasks

## Core Definitions of Reciprocal Trigonometric Functions

**Reciprocal Trigonometric Functions** — The three trigonometric functions defined as the multiplicative inverses of cosine, sine, and tangent respectively, for all inputs where the base function is non-zero.

*Notation:* $\sec \theta$, $\csc \theta$, $\cot \theta$

$$\sec \theta = \frac{1}{\cos \theta}, \quad \csc \theta = \frac{1}{\sin \theta}, \quad \cot \theta = \frac{\cos \theta}{\sin \theta}$$

Note that cotangent can be written both as $1/\tan \theta$ and the ratio of cosine to sine; the second form avoids undefined values when tangent itself is undefined, which simplifies asymptote calculations.

**Worked example:** Evaluate $\sec(60^\circ)$, $\csc(3\pi/2)$, and $\cot(5\pi/4)$ using unit circle values.

1. First retrieve the base sine and cosine values for each angle:

   $$\cos 60^\circ = 0.5, \quad \sin\left(\frac{3\pi}{2}\right) = -1, \quad \cos\left(\frac{5\pi}{4}\right) = \sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$$
2. Apply the reciprocal definitions:

   $$\sec 60^\circ = \frac{1}{0.5} = 2, \quad \csc\left(\frac{3\pi}{2}\right) = \frac{1}{-1} = -1, \quad \cot\left(\frac{5\pi}{4}\right) = \frac{-\sqrt{2}/2}{-\sqrt{2}/2} = 1$$

> **Exam tip:** The AP exam almost never asks you to compute these values directly, but you will need them as intermediate steps for larger problems.

## Domain, Range, and Asymptote Properties

Since each reciprocal function is undefined when its base sine or cosine is zero, we can derive vertical asymptote locations directly from the zeros of the base function.

| Function | Undefined When | Vertical Asymptotes (radians) | Domain | Range |
| --- | --- | --- | --- | --- |
| $\sec \theta$ | $\cos \theta = 0$ | $\theta = \frac{\pi}{2} + k\pi$ for all integers $k$ | All real numbers except $\frac{\pi}{2} + k\pi$ | $(-\infty, -1] \cup [1, \infty)$ |
| $\csc \theta$ | $\sin \theta = 0$ | $\theta = k\pi$ for all integers $k$ | All real numbers except $k\pi$ | $(-\infty, -1] \cup [1, \infty)$ |
| $\cot \theta$ | $\sin \theta = 0$ | $\theta = k\pi$ for all integers $k$ | All real numbers except $k\pi$ | All real numbers |

> **info**
>
> Notice that secant and cosecant never produce output values between -1 and 1, because the absolute value of cosine and sine is always at most 1, so their reciprocals have absolute value at minimum 1.

**Check your understanding**

Test your understanding of domain rules:

1. Which of the following angles is not in the domain of $\csc \theta$?

   - A. $\pi/3$
   - B. $\pi/2$
   - C. $\pi$
   - D. $3\pi/2$

   *Why:* $\sin \pi = 0$, so $\csc \pi = 1/0$ which is undefined, so $\pi$ is excluded from the domain of cosecant.

## Graphs of Secant, Cosecant, and Cotangent

To sketch a secant or cosecant graph, first draw the corresponding cosine or sine wave, then draw vertical asymptotes at every zero of that wave, then draw U-shaped reciprocal curves between each pair of asymptotes touching the peaks and troughs of the base sine/cosine graph.

**Derivation:** Derive the period of $\cot \theta$

*Starting from:* $\tan \theta$ has period $\pi$, and $\cot \theta = -\tan(\theta - \pi/2)$

1. A horizontal shift and vertical reflection cannot change the period of a periodic function
2. Since tangent repeats every $\pi$ radians, cotangent must also repeat every $\pi$ radians

*Conclusion:* The period of untransformed $\cot \theta$, $\sec \theta$, and $\csc \theta$ is $\pi$, $2\pi$, and $2\pi$ respectively.

**Worked example:** Find the period and locations of two consecutive vertical asymptotes for $f(x) = 2 \csc(3x)$.

1. First, identify the transformation applied to the base cosecant function. The horizontal scaling factor is 3, so the period is:

   $$T = \frac{2\pi}{3}$$
2. Base cosecant has asymptotes at $0$ and $\pi$, so scale these by $1/3$ to get the new asymptote locations:

   $$x=0, \quad x = \frac{\pi}{3}$$

**Exam command terms**

AP exam questions use specific command terms for graphing tasks:

- **Identify key features** — You only need to list asymptotes, period, and range, no full sketch required

- **Sketch the function** — You must label at least 2 asymptotes and 2 reference points on the graph to earn full credit

## Reciprocal Pythagorean Identities

**Derivation:** Derive the Pythagorean identity for secant and tangent

*Starting from:* Core Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$

1. Divide every term in the identity by $\cos^2 \theta$
2. Simplify each term: $\frac{\sin^2 \theta}{\cos^2 \theta} = \tan^2 \theta$, $\frac{\cos^2 \theta}{\cos^2 \theta} = 1$, $\frac{1}{\cos^2 \theta} = \sec^2 \theta$

*Conclusion:* The resulting identity is $1 + \tan^2 \theta = \sec^2 \theta$

$$1 + \cot^2 \theta = \csc^2 \theta$$

> **mnemonic**
>
> To remember which identity pairs with which function: the pair with tangent and secant uses 1 plus tan squared, the pair with cotangent and cosecant uses 1 plus cot squared, matching the order of the 6 trig functions.

## Common pitfalls

- **Wrong:** Defining cotangent as $1/\tan \theta$ and then incorrectly evaluating it at $\theta = \pi/2$ where tangent is undefined
  - Why it fails: At $\theta = \pi/2$, $\tan \theta$ is undefined, but $\cot \theta = \cos \theta / \sin \theta = 0$, so the reciprocal of tangent form breaks here
  - Correct: Always use $\cot \theta = \cos \theta / \sin \theta$ for all calculations to avoid missing valid defined values.
- **Wrong:** Thinking the range of secant includes values between -1 and 1
  - Why it fails: Students confuse the range of cosine (which is [-1,1]) with its reciprocal
  - Correct: Remember that if $|x| \leq 1$, then $|1/x| \geq 1$ for non-zero $x$.
- **Wrong:** Assigning a period of $2\pi$ to the cotangent function
  - Why it fails: Students assume all trigonometric functions have period $2\pi$
  - Correct: Memorize that only tangent and cotangent have base period $\pi$, the other four trig functions have base period $2\pi$.
- **Wrong:** Drawing secant graphs that cross the x-axis
  - Why it fails: Secant is 1/cosine, which can never equal zero, since the numerator is always 1
  - Correct: Reciprocal functions of sine and cosine never have x-intercepts, only vertical asymptotes.
- **Wrong:** Forgetting to scale asymptote locations when applying horizontal stretches/compressions
  - Why it fails: Students leave asymptotes at the base function positions instead of adjusting for the B value in $A\sec(Bx)$
  - Correct: Multiply all base asymptote x-values by $1/B$ to get the new positions after horizontal scaling.

## Cheatsheet

| Function | Reciprocal Definition | Base Period | Asymptote Locations | Range |
| --- | --- | --- | --- | --- |
| $\sec \theta$ | $1/\cos \theta$ | $2\pi$ | $\pi/2 + k\pi$ | $(-\infty,-1] \cup [1,\infty)$ |
| $\csc \theta$ | $1/\sin \theta$ | $2\pi$ | $k\pi$ | $(-\infty,-1] \cup [1,\infty)$ |
| $\cot \theta$ | $\cos \theta / \sin \theta$ | $\pi$ | $k\pi$ | $\mathbb{R}$ |
| Identity 1 | $1 + \tan^2 \theta = \sec^2 \theta$ | - | - | - |
| Identity 2 | $1 + \cot^2 \theta = \csc^2 \theta$ | - | - | - |

## What's next

Mastering these reciprocal trigonometric functions is a critical prerequisite for upcoming AP Precalculus topics, including solving trigonometric equations, verifying trigonometric identities, and working with polar coordinates. You will use secant, cosecant, and cotangent regularly in both the multiple choice and free response sections of the AP exam, especially when working with right triangle applications and periodic function modeling. These functions also lay groundwork that you will build on in future calculus courses.

- [Trigonometric Identity Verification](https://www.owlsprep.com/study/ap-precalculus-u3-trigonometric-identities/)

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