Overview

Inverse trig functions return principal values. Many contest errors come from forgetting these restricted ranges.

Principal Ranges

• $\arcsin x \in [-\pi/2, \pi/2]$.
• $\arccos x \in [0, \pi]$.
• $\arctan x \in (-\pi/2, \pi/2)$.

Core Skills

Use Principal Ranges

Always map the result back into the correct range for the inverse function.

Simplify Compositions

For $\arcsin(\sin x)$ and $\arccos(\cos x)$, reduce $x$ to the principal range first.

Convert Units

Keep angles in degrees or radians consistently; convert early if needed.

Compositions

• $\sin(\arcsin x)=x$ for $|x|\le 1$.
• $\arcsin(\sin x)$ equals $x$ only if $x$ is in $[-\pi/2,\pi/2]$.

Worked Example

Evaluate $\arcsin(\sin 200^\circ)$.

$\sin 200^\circ = -\sin 20^\circ$, so the principal value is $-20^\circ$.

More Examples

Example 1: Arccos Composition

Evaluate $\arccos(\cos 240^\circ)$.

$\cos 240^\circ = -1/2$, so the principal value is $120^\circ$.

Example 2: Arctan Range

Evaluate $\arctan(\tan 150^\circ)$.

$\tan 150^\circ = -1/\sqrt{3}$, so the principal value is $-30^\circ$.

Example 3: Radian Form

Evaluate $\arcsin(\sin(5\pi/6))$.

$\sin(5\pi/6)=1/2$, so the principal value is $\pi/6$.

Strategy Checklist

• Identify the principal range of the inverse function.
• Reduce the angle to that range.
• Keep degree/radian units consistent.

Common Pitfalls

• Assuming $\arcsin(\sin x)=x$ for all $x$.
• Using degrees for some steps and radians for others.
• Returning an angle outside the principal range.

Practice Problems