Find the Exact Value 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. Relate the trigonometric 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)

(4/5)2+cos2(θ)=1

4. Simplify the squared fraction and subtract it from both sides.

16/25+cos2(θ)=1

cos2(θ)=1−16/25

cos2(θ)=9/25

5. Solve for cos(θ) by taking the square root. Since θ=arcsin(4/5) is in the first quadrant (where sine is positive), cos(θ) must be positive.

cos(θ)=√(,9/25)

cos(θ)=3/5

Final Answer

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

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