Finding cos(arcsin(x))

Trigonometry · Axiom Academy

EXAMPLE Finding cos(arcsin(x)) Derive a general formula using the Pythagorean identity and express the answer algebraically in terms of x Excellent work! You've successfully derived the formula for cos(arcsin(x)). Here's what we learned: Inverse Function Strategy: Let θ = arcsin(x), which means sin(θ) = x. This converts the problem into finding cos(θ). Geometric Interpretation: Visualize sin(θ) = x as a right triangle with opposite side x and hypotenuse 1. Pythagorean Identity: Use x² + (adjacent)² = 1² to find the adjacent side: √(1 - x²). Final Formula: cos(arcsin(x)) = √(1 - x²), valid for -1 ≤ x ≤ 1. Domain Restriction: The expression is only defined when |x| ≤ 1, which matches the domain of arcsin. This technique works for other inverse trig compositions too! Try applying the same method to find sin(arccos(x)) or tan(arcsin(x)).

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