Simplify cos(arcsin(4/5))

Problem

cos(arcsin(4/5))

Solution

1. Identify the inner expression as an angle θ=arcsin(4/5) which implies sin(θ)=4/5 where θ is in the interval [−π/2,π/2]

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

3. Substitute the known value of sin(θ) into the identity: (4/5)2+cos2(θ)=1

4. Solve for cos2(θ) by calculating 1 - 16/25 = 9/25$.

5. Determine the sign of cos(θ) by noting that for θ=arcsin(4/5) the angle is in the first quadrant, so cos(θ) must be positive.

6. Calculate the square root: cos(θ)=√(,9/25)=3/5

Final Answer

cos(arcsin(4/5))=3/5

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