Simplify sin(arccos(x))

Problem

sin(arccos(x))

Solution

1. Identify the relationship between the functions by letting θ=arccos(x)

2. Apply the definition of the inverse cosine function, which implies cos(θ)=x for 0≤θ≤π

3. Use the Pythagorean identity sin2(θ)+cos2(θ)=1 to relate the sine and cosine of the angle.

4. Solve for sine by substituting x for cos(θ) resulting in sin2(θ)+x2=1

5. Isolate the sine term to get sin2(θ)=1−x2

6. Take the square root of both sides, noting that since 0≤θ≤π sin(θ) must be non-negative, so sin(θ)=√(,1−x2)

Final Answer

sin(arccos(x))=√(,1−x2)

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