Defining Arcsin, Arccos, and Arctan

Trigonometry · Axiom Academy

LESSON Inverse Trigonometric Functions Understanding arcsin, arccos, and arctan: definitions, restricted domains, and the inverse relationships with their parent functions. 1. The Inverse Function Problem To create an inverse function, the original function must be one-to-one (each output corresponds to exactly one input). Let's see why sine poses a problem: 2. The Solution: Restrict the Domain To create proper inverse functions, we restrict each trigonometric function to a domain where it's one-to-one. We choose intervals that: Cover the full range of outputs Include familiar angles like 0, π/2, etc. The domain of arcsin is [-1, 1] (all possible sine values). The range of arcsin is [-π/2, π/2] (restricted sine domain). 4. Arccosine (arccos or cos⁻¹) The domain of arccos is [-1, 1] (all possible cosine values). The range of arccos is [0, π] (restricted cosine domain). 5. Arctangent (arctan or tan⁻¹) The domain of arctan is (-∞, ∞) (all real numbers). The range of arctan is (-π/2, π/2) (restricted tangent domain). Here's a quick reference comparing all three inverse trigonometric functions: sin(arcsin(x)) = x for x ∈ [-1, 1] arcsin(sin(x)) = x for x ∈ [-π/2, π/2] cos(arccos(x)) = x for x ∈ [-1, 1] arccos(cos(x)) = x for x ∈ [0, π] tan(arctan(x)) = x for all x ∈ ℝ arctan(tan(x)) = x for x ∈ (-π/2, π/2)

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