Reciprocal trig functions (sec, csc, cot)
Reciprocal Trigonometric Functions
Reciprocal Trigonometric Functions is a topic in Trigonometric Functions in the Texas Essential Knowledge and Skills (Precalculus, §111.42). It is aligned to Standard P.2(F), which requires students to graph the secant, cosecant, and cotangent 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.
Theory
Each basic trig function has a reciprocal:
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.
The reciprocal identities:
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.
Secant is the reciprocal of cosine.
| \(\sec 60^\circ\) | \(=\) | \(\dfrac{1}{\cos 60^\circ}\) |
| \(=\) | \(\dfrac{1}{1/2}=2\) |
Cotangent is cosine over sine (the reciprocal of tangent).
| \(\cot\dfrac{\pi}{4}\) | \(=\) | \(\dfrac{\cos(\pi/4)}{\sin(\pi/4)}\) |
| \(=\) | \(\dfrac{\sqrt2/2}{\sqrt2/2}=1\) |
\(\csc x=\dfrac{1}{\sin x}\) is undefined where \(\sin x=0\).
| \(\sin x=0\) | \(\Rightarrow\) | \(x=\pi n\) |
Vertical asymptotes at every integer multiple of \(\pi\).
Since \(\csc x=\dfrac{1}{\sin x}\), the product collapses.
| \(\sin x\cdot\csc x\) | \(=\) | \(\sin x\cdot\dfrac{1}{\sin x}\) |
| \(=\) | \(1\) |
Common pitfalls
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.