Pre-Calculus Trigonometric functions

Reciprocal trig functions (sec, csc, cot)

Q: What are the reciprocal trig functions?

Cosecant is 1 over sine, secant is 1 over cosine, and cotangent is cosine over sine. They are the reciprocals of sine, cosine, and tangent.

Q: Is secant the reciprocal of sine or cosine?

Of cosine: secant is 1 over cosine. Cosecant is the reciprocal of sine. The names cross over.

Q: Where are the reciprocal functions undefined?

Where their denominators are zero: cosecant and cotangent where sine is zero, and secant where cosine is zero.

Q: Is sec x the same as arccos x?

No. Secant x is the reciprocal 1 over cosine x; arccosine is the inverse function. They are different.

20 practice questions 0 video lessons Theory + worked examples

Reciprocal Trigonometric Functions

California Pre-Calculus • Standard F-IF.7e • Trigonometric Functions

Reciprocal Trigonometric Functions is a topic in Trigonometric Functions in the California Common Core State Standards. It is aligned to Standard F-IF.7e, which requires students to graph the reciprocal trigonometric functions.

The reciprocal trigonometric functions are \(\csc x=\dfrac{1}{\sin x}\), \(\sec x=\dfrac{1}{\cos x}\), and \(\cot x=\dfrac{1}{\tan x}\), with asymptotes where the base function is zero.

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Theory

Each basic trig function has a reciprocal:

\[\csc x=\dfrac{1}{\sin x},\qquad \sec x=\dfrac{1}{\cos x},\qquad \cot x=\dfrac{1}{\tan x}=\dfrac{\cos x}{\sin x}.\]

Their graphs have vertical asymptotes wherever the original function is zero (you can't divide by zero), and they touch the original curve at its maxima and minima.

Mind the pairing: secant goes with cosine (both have the “co” mismatch), and cosecant goes with sine. It's the opposite of what the names suggest.

\(\sec x=\dfrac{1}{\cos x}\): asymptotes where \(\cos x=0\).

\(\csc x=\dfrac{1}{\sin x}\): asymptotes where \(\sin x=0\).

The reciprocal identities:

\[\csc x=\dfrac{1}{\sin x},\quad \sec x=\dfrac{1}{\cos x},\quad \cot x=\dfrac{\cos x}{\sin x}\]

cosecant is 1 over sine, secant is 1 over cosine, cotangent is cosine over sine

Undefined where the denominator is zero: \(\csc,\cot\) at \(\sin x=0\); \(\sec\) at \(\cos x=0\).

How to work with reciprocal functions

To evaluate: compute the basic function first, then take the reciprocal.
To locate asymptotes: set the underlying sine or cosine to zero.
To simplify: rewrite everything in terms of \(\sin\) and \(\cos\), then cancel.

Example 1 — Evaluate secant

Find \(\sec 60^\circ\).

Solution

Secant is the reciprocal of cosine.

\(\sec 60^\circ\)	\(=\)	\(\dfrac{1}{\cos 60^\circ}\)
	\(=\)	\(\dfrac{1}{1/2}=2\)

secant of 60 degrees is 2

Example 2 — Evaluate cotangent

Find \(\cot\dfrac{\pi}{4}\).

Solution

Cotangent is cosine over sine (the reciprocal of tangent).

\(\cot\dfrac{\pi}{4}\)	\(=\)	\(\dfrac{\cos(\pi/4)}{\sin(\pi/4)}\)
	\(=\)	\(\dfrac{\sqrt2/2}{\sqrt2/2}=1\)

cotangent of pi over 4 is 1

Example 3 — Where cosecant is undefined

Where is \(\csc x\) undefined?

Solution

\(\csc x=\dfrac{1}{\sin x}\) is undefined where \(\sin x=0\).

\(\sin x=0\)

\(\Rightarrow\)

\(x=\pi n\)

Vertical asymptotes at every integer multiple of \(\pi\).

cosecant undefined at integer multiples of pi

Example 4 — Simplify with reciprocals

Simplify \(\sin x\cdot\csc x\).

Solution

Since \(\csc x=\dfrac{1}{\sin x}\), the product collapses.

\(\sin x\cdot\csc x\)	\(=\)	\(\sin x\cdot\dfrac{1}{\sin x}\)
	\(=\)	\(1\)

sine times cosecant equals 1

Common pitfalls

\(\sec\) pairs with \(\cos\), \(\csc\) with \(\sin\). The names cross over — a frequent mix-up.

Reciprocal, not inverse. \(\sec x=\dfrac{1}{\cos x}\), which is not \(\cos^{-1}x\) (arccosine).

Undefined at the zeros of the base function. There the reciprocal has an asymptote.

Frequently asked questions

What are the reciprocal trig functions?

Cosecant \(=1/\sin\), secant \(=1/\cos\), and cotangent \(=\cos/\sin\). They are the reciprocals of sine, cosine, and tangent.

Is secant the reciprocal of sine or cosine?

Of cosine: \(\sec x=1/\cos x\). Cosecant is the reciprocal of sine. The names cross over.

Where are the reciprocal functions undefined?

Where their denominators are zero: \(\csc\) and \(\cot\) at \(\sin x=0\), and \(\sec\) at \(\cos x=0\).

Is sec x the same as arccos x?

No. \(\sec x\) is the reciprocal \(1/\cos x\); \(\cos^{-1}x\) (arccosine) is the inverse function. They are different.

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Graphs of sine, cosine, and tangent

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Amplitude, period, phase shift, and midline

Reciprocal trig functions (sec, csc, cot)

Reciprocal Trigonometric Functions

📖 Theory

How to work with reciprocal functions

Common pitfalls

Frequently asked questions

Theory