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

3. Solve for sin(θ) by substituting u for cos(θ) resulting in sin2(θ)+u2=1

4. Isolate sin(θ) by subtracting u2 from both sides to get sin2(θ)=1−u2

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−u2)

Final Answer

sin(arccos(u))=√(,1−u2)

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