Reciprocal Functions and Inverse Functions

Two different ideas share the same −1 symbol, and that is where most of the confusion in trigonometry starts. A reciprocal flips a value; an inverse reverses the whole operation.

Concept What goes in, what comes out

Reciprocal functions — $\csc\theta = \dfrac{1}{\sin\theta}$ , $\sec\theta = \dfrac{1}{\cos\theta}$ , $\cot\theta = \dfrac{1}{\tan\theta}$ — answer "what is the flipped value of this ratio?" The input is an angle and the answer is a number.

Inverse functions — $\arcsin$ , $\arccos$ , $\arctan$ — answer "which angle produced this ratio?" The input is a number and the answer is an angle.

Note One symbol, two meanings

\sin^{-1}\!\left(\tfrac{1}{2}\right) = 30°

— the inverse; the answer is an angle.

(\sin 30°)^{-1} = 2

— the reciprocal; the answer is a number.

The position of the −1 decides everything. Writing $\arcsin$ instead of $\sin^{-1}$ removes the ambiguity.

Example Reciprocals

\sin 30° = \tfrac{1}{2}

, so

\csc 30° = \dfrac{1}{1/2}

⟹ csc 30° = 2

\cos 60° = \tfrac{1}{2}

, so

\sec 60° = \dfrac{1}{1/2}

⟹ sec 60° = 2

\tan 45° = 1

, so

\cot 45° = \dfrac{1}{1}

⟹ cot 45° = 1

Example Inverses

Which angle has a sine of

\tfrac{1}{2}

\arcsin\!\left(\tfrac{1}{2}\right)

⟹ 30°

Which angle has a cosine of

\dfrac{\sqrt{3}}{2}

\arccos\!\left(\dfrac{\sqrt{3}}{2}\right)

⟹ 30°

Which angle has a tangent of 1?

\arctan(1)

⟹ 45°

Note Two common mistakes

Believing that $\sin^{-1}x = \dfrac{1}{\sin x}$ . It does not: $\sin^{-1}x = \arcsin x$ is the inverse, while $\dfrac{1}{\sin x} = \csc x$ is the reciprocal.

Writing $\arcsin(2)$ and expecting an answer. The input 2 is outside the domain −1 to 1, so the value does not exist. In the same way $\csc 0°$ fails, because $\sin 0° = 0$ .

Reference Side by side

Property	Reciprocal functions	Inverse functions
Symbols	csc, sec, cot	arcsin, arccos, arctan
Input	an angle θ	a number x
Output	a number	an angle
Example	csc 30° = 2	arcsin(½) = 30°
Range	csc, sec: ≤ −1 or ≥ 1; cot: all reals	arcsin −90°…90°; arccos 0°…180°; arctan between ±90°

Summary

Reciprocal functions flip the ratio: angle in, number out.
Inverse functions reverse the operation: number in, angle out.
sin⁻¹x means the inverse; (sin x)⁻¹ means the reciprocal — the position of the −1 decides.
Use arc notation for inverses, and check the domain before evaluating.