Question

When you use Inverse Trigonometry, what are you looking for?

Answer 1

You're looking for an angle.

Inverse trig functions (arcsin, arccos, arctan, etc.) answer the question: "What angle has this sine/cosine/tangent value?" For example:
- arcsin(1/2) = 30° (π/6) because sin(30°) = 1/2.
- arctan(1) = 45° (π/4) because tan(45°) = 1.

Notes you should know:
- Calculators return a principal value (restricted range): arcsin ∈ [−π/2, π/2], arccos ∈ [0, π], arctan ∈ (−π/2, π/2). Other angles with the same trig value may exist.
- For a full set of solutions on the unit circle:
- If sin θ = x: θ = arcsin(x) + 2πk or θ = π − arcsin(x) + 2πk.
- If cos θ = x: θ = ±arccos(x) + 2πk (equivalently θ = arccos(x) + 2πk or θ = −arccos(x) + 2πk).
- If tan θ = x: θ = arctan(x) + πk.
- In right-triangle problems, use inverse trig to find an angle from a side ratio: θ = arcsin(opposite/hypotenuse), θ = arccos(adjacent/hypotenuse), θ = arctan(opposite/adjacent).

That’s the idea: you convert a known trig ratio back into the corresponding angle (keeping in mind principal values and additional solutions).