Graphs of Cosecant, Secant, and Cotangent
Simple Explanation
The remaining three trigonometric functions are defined as reciprocals of the first three: csc x = 1/sin x, sec x = 1/cos x, and cot x = 1/tan x. Each has its own vertical asymptotes, occurring wherever its reciprocal partner equals zero.
Why Do We Need It?
These reciprocal functions round out the full family of six trigonometric functions, each useful in different problem contexts.
Formula
The Reciprocal Trigonometric Functions
csc x = 1/sin x, sec x = 1/cos x, cot x = 1/tan x
Cosecant, secant, and cotangent are defined as the reciprocals of sine, cosine, and tangent respectively.
- csc, sec, cot
- β the reciprocal trigonometric functions
When to use it: Whenever cosecant, secant, or cotangent needs to be evaluated or graphed.
Worked Example
Evaluate a reciprocal trig function and explain an undefined case
Find sec(0), and explain why csc(0) is undefined.
Why Does This Work?
Since each of these functions is DEFINED as a reciprocal (1 divided by another trig function), it is undefined precisely wherever that other function equals zero β exactly like 1/0 is always undefined.
Real-Life Example
Optics and lens calculations
Certain optical formulas involving angles of incidence are naturally expressed using secant or cosecant rather than cosine or sine directly.
Working with the reciprocal functions can simplify these particular optical equations.
Practice
Find cot(Ο/4).
MediumCommon mistake
Assuming a reciprocal function is undefined at the same x-values as its partner β e.g. csc is undefined where SIN is zero, not where sin is itself undefined (sine is never undefined).
Quick Review
- csc x = 1/sin x, sec x = 1/cos x, cot x = 1/tan x.
- Each is undefined exactly where its reciprocal partner equals zero.
- These round out the full family of six trigonometric functions.