Expand the Trigonometric Expression sin(arccos(2x))

Problem

sin(arccos(2*x))

Solution

1. Identify the inner function as an angle θ=arccos(2*x) which implies cos(θ)=2*x where 0≤θ≤π

2. Apply the Pythagorean identity sin2(θ)+cos2(θ)=1 to relate the sine and cosine functions.

3. Solve for sin(θ) by rearranging the identity to sin(θ)=±√(,1−cos2(θ))

4. Determine the sign of the square root by noting that for the range of the arccosine function [0,π] the sine function is always non-negative, so sin(θ)=√(,1−cos2(θ))

5. Substitute the value cos(θ)=2*x into the expression to get √(,1−(2*x)2)

6. Simplify the expression inside the square root.

Final Answer

sin(arccos(2*x))=√(,1−4*x2)

Want more problems? Check here!