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

Secant, cosecant & cotangent

What is the secant function?

• The secant function is the reciprocal of the cosine function

  • $\sec \theta =\frac{1}{\cos \theta },\text{where }\cos \theta \ne 0$

• The secant function is undefined wherever $\cos \theta =0$

  • i.e. at $\theta =\frac{\pi }{2}+k\pi$ for integer values of $k$

What is the cosecant function?

• The cosecant function is the reciprocal of the sine function

  • $\csc \theta =\frac{1}{\sin \theta },\text{where }\sin \theta \ne 0$

• The cosecant function is undefined wherever $\sin \theta =0$

  • i.e. at $\theta =k\pi$ 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

  • $\sec \theta$ has vertical asymptotes at $\theta =\frac{\pi }{2}+k\pi$

    • the same values where $\cos \theta =0$

  • $\csc \theta$ has vertical asymptotes at $\theta =k\pi$

    • the same values where $\sin \theta =0$

• The range of both secant and cosecant is $(-\infty , -1]\cup [1, \infty )$

  • Output values are always $\le -1$ or $\ge 1$

    • 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 $2\pi$ (same as cosine)

  • and cosecant has period $2\pi$ (same as sine)

What is the cotangent function?

• The cotangent function is the reciprocal of the tangent function

  • $\cot \theta =\frac{1}{\tan \theta },\text{where }\tan \theta \ne 0$

• Equivalently, cotangent can be written as the ratio of cosine to sine

  • $\cot \theta =\frac{\cos \theta }{\sin \theta },\text{where }\sin \theta \ne 0$

• The cotangent function is undefined wherever $\sin \theta =0$

  • i.e. at $\theta =k\pi$ for integer values of $k$

What are the key characteristics of the cotangent graph?

• The graph of cotangent has vertical asymptotes at values where $\tan \theta =0$

  • i.e. at $\theta =k\pi$ for integer values of $k$

  • Note that these are the same values where $\sin \theta =0$

    • which is consistent with the formula $\cot \theta =\frac{\cos \theta }{\sin \theta }$

• 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 $\pi$ (the same as tangent)

• The range of cotangent is all real numbers (the same as tangent)

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:

  • $\sec \frac{\pi }{3}=\frac{1}{\cos \frac{\pi }{3}}=\frac{1}{\frac{1}{2}}=2$

  • $\csc \frac{\pi }{4}=\frac{1}{\sin \frac{\pi }{4}}=\frac{1}{\frac{\sqrt{2}}{2}}=\frac{2}{\sqrt{2}}=\sqrt{2}$

  • $\cot \frac{\pi }{6}=\frac{1}{\tan \frac{\pi }{6}}=\frac{1}{\frac{\sqrt{3}}{3}}=\frac{3}{\sqrt{3}}=\sqrt{3}$

• If the base function equals zero at that angle, the reciprocal function is undefined

  • E.g. $\csc \pi$ is undefined because $\sin \pi =0$