Find the Exact Value cos(arcsin(4/9))

Problem

cos(arcsin(4/9))

Solution

1. Identify the inner expression as an angle θ=arcsin(4/9) which implies sin(θ)=4/9 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. Solve for cos(θ) by rearranging the identity to cos(θ)=√(,1−sin2(θ)) noting that cos(θ) must be positive because θ is in the first quadrant (since 4/9>0.

4. Substitute the value sin(θ)=4/9 into the equation.

cos(θ)=√(,1−(4/9)2)

5. Simplify the expression inside the square root.

cos(θ)=√(,1−16/81)

cos(θ)=√(,81/81−16/81)

cos(θ)=√(,65/81)

6. Evaluate the square root of the numerator and denominator.

cos(θ)=√(,65)/9

Final Answer

cos(arcsin(4/9))=√(,65)/9

Want more problems? Check here!