Simplify sin(arccos(1-x))

Problem

sin(arccos(1−x))

Solution

1. Identify the relationship between the trigonometric functions. Let θ=arccos(1−x) which implies cos(θ)=1−x where 0≤θ≤π

2. Apply the Pythagorean identity sin2(θ)+cos2(θ)=1 to find sin(θ)

3. Solve for sin(θ) by substituting the expression for cos(θ)

sin(θ)=√(,1−cos2(θ))

4. Substitute 1−x for cos(θ)

sin(arccos(1−x))=√(,1−(1−x)2)

5. Expand the squared binomial (1−x)2

(1−x)2=1−2*x+x2

6. Simplify the expression inside the square root.

1−(1−2*x+x2)=1−1+2*x−x2

2*x−x2

Final Answer

sin(arccos(1−x))=√(,2*x−x2)

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