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Inverse Functions

The Hook

The Undo Button.

What is it?

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

Time Travel (History)

developed alongside standard trig to reverse-engineer astronomical observations.

Cheat Code

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

Practice This Now!

The Field Guide

In 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?'

In The Real World: 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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Booby Trap!

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

Real World Challenge
"A road rises 10 meters for every 100 meters of horizontal distance. What is the angle of the incline?"
The Logic: 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.

Training Drills

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