Inverse Trigonometry
Inverse trigonometric functions — arcsin, arccos, and arctan — determine the angle that produces a specific trigonometric ratio. Unlike regular trigonometric functions that map angles to ratios, inverse functions map ratios back to angles within restricted ranges. The principal ranges ensure each input has exactly one output: arcsin operates on [−π/2, π/2], arccos on [0, π], and arctan on (−π/2, π/2).
Why it matters
Inverse trigonometric functions solve real-world problems requiring angle calculations from known measurements. Engineers use arctan to determine roof slopes when the rise-to-run ratio is 3:4, yielding an angle of approximately 36.87°. GPS navigation systems employ arcsin and arccos to calculate bearing angles between coordinates. In physics, projectile motion problems use arcsin to find launch angles — a ball thrown at 20 m/s reaching maximum range requires arcsin(1) ÷ 2 = 45°. Computer graphics rely on inverse trig to rotate objects through specific angles determined by coordinate ratios. These functions appear throughout A-level Further Mathematics and university calculus courses, particularly in integration techniques and differential equations.
How to solve inverse trigonometry
Inverse Trig — arcsin, arccos, arctan
- Read arcsin(v) as 'the angle whose sine is v'.
- Principal ranges: arcsin ∈ [−π/2, π/2], arccos ∈ [0, π], arctan ∈ (−π/2, π/2).
- Use unit-circle values in reverse to evaluate at standard inputs.
- For compositions like sin(arccos(v)): let θ = arccos(v), then use sin²θ + cos²θ = 1.
Example: arcsin(12) = π/6. sin(arccos(12)) = sin(π/3) = √32.
Worked examples
Find the exact value of arccos(1) in degrees.
Answer: 0°
- Ask: what angle has cosine equal to 1? → arccos(1) = 0° — Inverse trig undoes the regular function. You read it as 'the angle whose cosine is 1'. Use your memorised unit-circle values to find the matching angle.
Find the exact value of arccos(−1) in radians.
Answer: π
- Find the angle whose cos is −1, respecting the principal range → arccos(−1) = π — arccos has a restricted range so that every input has exactly one output. Pick the angle within that range.
Evaluate arcsin(0) and explain why this is the only valid answer.
Answer: 0
- List all angles that satisfy the inner equation → multiple angles from periodicity — Periodic functions have infinitely many solutions; the inverse must pick one.
- Restrict to the principal range [−π/2, π/2] → arcsin(0) = 0 — sin x = 0 at x = 0, π, 2π, −π, ... Only x = 0 lies in [−π/2, π/2], so arcsin(0) = 0.
Common mistakes
- Confusing degrees and radians leads to errors like writing arcsin(1/2) = 30 instead of π/6 when working in radians.
- Ignoring principal ranges causes mistakes such as claiming arccos(−1/2) = −π/3 instead of the correct answer 2π/3.
- Mishandling compositions results in errors like sin(arccos(1/2)) = 1/2 instead of the correct value √3/2.