Evaluate 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 where θ is in the interval [−π/2,π/2]

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

3. Substitute the known value of sin(θ) into the identity: cos2(θ)+(5/13)2=1

4. Solve for cos2(θ) by calculating cos2(θ)=1−25/169

5. Simplify the fraction to get cos2(θ)=144/169

6. Determine the sign of cos(θ) by noting that for θ=arcsin(5/13) the angle is in the first quadrant where cosine is positive.

7. Calculate the square root to find cos(θ)=12/13

Final Answer

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

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