Simplify cos(arcsin(5/13))

Problem

cos(arcsin(5/13))

Solution

1. Identify the inner expression as an angle θ=arcsin(5/13) which implies sin(θ)=5/13 for an angle in the interval [−π/2,π/2]

2. Relate the sine and cosine functions using the Pythagorean identity sin2(θ)+cos2(θ)=1

3. Substitute the known value of sin(θ) into the identity to solve for cos(θ)

(5/13)2+cos2(θ)=1

4. Calculate the square of the fraction.

25/169+cos2(θ)=1

5. Isolate the cosine term by subtracting the fraction from both sides.

cos2(θ)=1−25/169

cos2(θ)=169/169−25/169

cos2(θ)=144/169

6. Solve for cos(θ) by taking the square root. Since θ=arcsin(5/13) is in the first quadrant, the cosine must be positive.

cos(θ)=√(,144/169)

cos(θ)=12/13

Final Answer

cos(arcsin(5/13))=12/13

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