Simplify cos(arcsin(( square root of 2)/3))

Problem

cos(arcsin(√(,2)/3))

Solution

1. Identify the inner expression as an angle θ where θ=arcsin(√(,2)/3)

2. Apply the definition of the inverse sine function, which implies sin(θ)=√(,2)/3 for −π/2≤θ≤π/2

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

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

5. Solve for cos2(θ) by calculating the square: cos2(θ)+2/9=1

6. Subtract 2/9 from both sides: cos2(θ)=7/9

7. Take the square root of both sides, noting that cos(θ) must be positive because θ is in the range of arcsin() which is [−π/2,π/2] cos(θ)=√(,7/9)

8. Simplify the radical expression: cos(θ)=√(,7)/3

Final Answer

cos(arcsin(√(,2)/3))=√(,7)/3

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