Find the Exact Value cos(arcsin(7/8))

Problem

cos(arcsin(7/8))

Solution

1. Identify the inner expression as an angle θ=arcsin(7/8) which implies sin(θ)=7/8 where θ is in the interval [−π/2,π/2]

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

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

cos2(θ)+(7/8)2=1

4. Calculate the square of the fraction.

cos2(θ)+49/64=1

5. Isolate the cos2(θ) term by subtracting 49/64 from both sides.

cos2(θ)=1−49/64

cos2(θ)=64/64−49/64

cos2(θ)=15/64

6. Solve for cos(θ) by taking the square root. Since θ=arcsin(7/8) is in the first quadrant, cos(θ) must be positive.

cos(θ)=√(,15/64)

cos(θ)=√(,15)/8

Final Answer

cos(arcsin(7/8))=√(,15)/8

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