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 — , , — answer "what is the flipped value of this ratio?" The input is an angle and the answer is a number.

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

Note One symbol, two meanings

— the inverse; the answer is an angle.

— the reciprocal; the answer is a number.

The position of the −1 decides everything. Writing instead of removes the ambiguity.

Example Reciprocals

, so

⟹ csc 30° = 2

, so

⟹ sec 60° = 2

, so

⟹ cot 45° = 1

Example Inverses

Which angle has a sine of

⟹ 30°

Which angle has a cosine of

⟹ 30°

Which angle has a tangent of 1?

⟹ 45°

Note Two common mistakes

Believing that . It does not: is the inverse, while is the reciprocal.

Writing and expecting an answer. The input 2 is outside the domain −1 to 1, so the value does not exist. In the same way fails, because .

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.