Simplify sin(arccos(u))

Problem

sin(arccos(u))

Solution

1. Define an angle θ such that θ=arccos(u) This implies that cos(θ)=u where 0≤θ≤π

2. Use the Pythagorean identity sin^2(θ)+cos^2(θ)=1 to relate the sine and cosine functions.

3. Solve for sin(θ) by substituting u for cos(θ) resulting in sin^2(θ)+u^2=1

4. Isolate sin(θ) by subtracting u^2 from both sides to get sin^2(θ)=1−u^2

5. Take the square root of both sides. Since the range of arccos(u) is [0,π] the sine of any angle in this range is non-negative, so sin(θ)=√(,1−u^2)

Final Answer

sin(arccos(u))=√(,1−u^2)

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