Inverse Functions Summary

Trigonometry · Axiom Academy

SUMMARY Inverse Trigonometric Functions Let's review how inverse trigonometric functions reverse the process, finding angles from ratios with proper domain restrictions. What are Inverse Trig Functions? Purpose: Inverse trigonometric functions reverse the trigonometric process—they find angles when given ratios. Notation: Written as arcsin, arccos, arctan (or sin⁻¹, cos⁻¹, tan⁻¹). The "arc" prefix means "angle whose..." Key Question: If sin(θ) = 0.5, what is θ? Answer: θ = arcsin(0.5) = π/6 One-to-One Requirement: For a function to have an inverse, it must be one-to-one. Original trig functions repeat, so domains must be restricted. The Problem: sin(π/6) = 1/2, but also sin(5π/6) = 1/2 and sin(π/6 + 2πn) = 1/2 for any integer n. Which angle does arcsin(1/2) return? The Solution: Restrict the original function's domain so each output corresponds to exactly one input angle. Standard Convention: Choose restrictions that include all possible output values and are as simple as possible (usually centered around 0). Single-Valued Outputs: With restrictions, arcsin(1/2) has one answer: π/6, making inverse functions useful for solving equations. Example: Evaluating cos(arcsin(3/5)) Understand the question: Find the cosine of an angle whose sine is 3/5 Let θ = arcsin(3/5): This means sin(θ) = 3/5 and θ is in the range [-π/2, π/2] Draw a right triangle: Since sin(θ) = opposite/hypotenuse = 3/5, draw a triangle with opposite side 3 and hypotenuse 5

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