Inverse Function Method

Trigonometry · Axiom Academy

LESSON Inverse Function Method Using arcsin, arccos, and arctan to find principal values, then generating all solutions using periodicity 1. Principal Values: The First Answer Inverse trigonometric functions (arcsin, arccos, arctan) always return a single value from a restricted range. This is called the principal value . arcsin(x): Returns values in [-π/2, π/2] arccos(x): Returns values in [0, π] arctan(x): Returns values in (-π/2, π/2) 2. Using Periodicity to Find All Solutions Once we have the principal value, we use the periodicity of trig functions to find all other solutions. The key insight: if θ is a solution, then θ + (period) is also a solution! 3. Solution Patterns for Each Function Each inverse function has its own pattern for generating all solutions. Here's the complete picture: 4. Complete Example: Solve sin(x) = 0.5 Let's solve this step by step using the inverse function method: Find the principal value: x₁ = arcsin(0.5) = π/6 Find the second value in [0, 2π): x₂ = π - π/6 = 5π/6 Add the period for all solutions: x = π/6 + 2πk or x = 5π/6 + 2πk sin(π/6 + 2π) = sin(π/6) = 0.5 ✓ sin(5π/6 + 2π) = sin(5π/6) = 0.5 ✓

This is the written version of the interactive lesson above. See the full Trigonometry course.