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Inverse Functions

Overview

The Undo Button.

What is it?

Functions that calculate the unknown angle when the side ratios are known.

History

developed alongside standard trig to reverse-engineer astronomical observations.

Key Idea

If you need the Angle, use the Inverse (Arcsin, Arccos, Arctan).

Practice This Topic

Concept Guide

Plain English: Regular Trig takes an angle and gives you a ratio. Inverse Trig takes the ratio and tells you what the angle originally was. It's asking 'What angle created this side length?'

Real-world example: Game Development. When a character aims a gun at a target, the code knows the X/Y coordinates (sides), but needs Inverse Tangent to figure out the rotation angle for the character model.

How to do it

  1. Determine which two sides you know (Opp, Adj, or Hyp).
  2. Choose the matching SOH CAH TOA function.
  3. Write the equation: Sin(x) = Ratio.
  4. Use the inverse button on the calc: x = Sin^-1(Ratio).
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Common Pitfall

Confusing Sin^-1 (Inverse Sine) with 1/Sin (Cosecant). They are totally different things!

Word Problem
"A road rises 10 meters for every 100 meters of horizontal distance. What is the angle of the incline?"
Reasoning: We have Opposite (10) and Adjacent (100). TOA says Tan(x) = 10/100. Use Inverse Tan: x = Arctan(0.1). The angle is approx 5.7 degrees.

Practice Examples

Notation
-1 Sin (x) OR arcsin(x)
Both mean 'The angle whose Sine is x'.