Reciprocal Trigonometric Functions

Intuition and core definition

Cosecant and secant are reciprocals of sine and cosine: $\csc\theta=1/\sin\theta$ where $\sin\theta\ne0$, and $\sec\theta=1/\cos\theta$ where $\cos\theta\ne0$. Define $\cot\theta=\cos\theta/\sin\theta$ where $\sin\theta\ne0$; the form $\cot\theta=1/\tan\theta$ is valid only when both $\sin\theta$ and $\cos\theta$ are nonzero.

Notation, language, and conditions

Reciprocal notation is not inverse-function notation: $\sin^{-1}x$ usually means arcsine, while $(\sin\theta)^{-1}=\csc\theta$. On a right triangle, $\sec=hypotenuse/adjacent$, $\csc=hypotenuse/opposite$, and $\cot=adjacent/opposite$.

Why this idea matters

Reciprocal trigonometric functions convert nonzero sine, cosine, and tangent values into cosecant, secant, and cotangent while exposing new undefined points.

A dependable method

1. Identify the base trig ratio or unit-circle coordinate.
2. Check whether that value is zero.
3. Take its multiplicative reciprocal, preserving sign.
4. Simplify exact radicals or fractions.
5. Multiply reciprocal pairs and expect $1$.

Worked example

Common mistakes