Simplify cos(arcsin((x-h)/r))

Problem

cos(arcsin((x−h)/r))

Solution

1. Identify the inner expression as an angle θ=arcsin((x−h)/r) which implies sin(θ)=(x−h)/r

2. Relate the trigonometric functions using the fundamental identity cos2(θ)+sin2(θ)=1

3. Solve for cos(θ) by rearranging the identity to cos(θ)=√(,1−sin2(θ))

4. Substitute the value of sin(θ) into the equation to get cos(θ)=√(,1−((x−h)/r)2)

5. Simplify the expression by finding a common denominator inside the square root.

6. Distribute the square root to the numerator and denominator.

Final Answer

cos(arcsin((x−h)/r))=√(,r2−(x−h)2)/r

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