Find the Exact Value cos(arcsin(2/3))

Problem

cos(arcsin(2/3))

Solution

1. Identify the inner expression as an angle θ=arcsin(2/3) which implies sin(θ)=2/3 where θ is in the interval [−π/2,π/2]

2. Use the Pythagorean identity cos2(θ)+sin2(θ)=1 to relate the sine and cosine of the angle.

3. Substitute the known value of sin(θ) into the identity:

cos2(θ)+(2/3)2=1

4. Solve for cos2(θ) by squaring the fraction and subtracting it from 1:

cos2(θ)+4/9=1

cos2(θ)=1−4/9

cos2(θ)=5/9

5. Determine the sign of cos(θ) Since θ=arcsin(2/3) and the range of arcsin() is [−π/2,π/2] the angle is in Quadrant I or IV where cosine is always positive.

6. Calculate the square root to find the final value:

cos(θ)=√(,5/9)

cos(θ)=√(,5)/3

Final Answer

cos(arcsin(2/3))=√(,5)/3

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