Simplify cos(arcsin(x))

Problem

cos(arcsin(x))

Solution

1. Identify the inner function and let θ=arcsin(x) This implies that sin(θ)=x where θ is in the interval [−π/2,π/2]

2. Relate the trigonometric functions using the Pythagorean identity cos2(θ)+sin2(θ)=1

3. Solve for cos(θ) by rearranging the identity to cos(θ)=±√(,1−sin2(θ))

4. Determine the sign of the square root. Since θ is in the range of the arcsine function [−π/2,π/2] the cosine of θ must be non-negative. Therefore, cos(θ)=√(,1−sin2(θ))

5. Substitute the value sin(θ)=x back into the expression to get √(,1−x2)

Final Answer

cos(arcsin(x))=√(,1−x2)

Want more problems? Check here!