Find the Exact Value sin(arccos(-2/3))

Problem

sin(arccos(−2/3))

Solution

1. Identify the inner function as an angle θ=arccos(−2/3) By the definition of the inverse cosine function, this means cos(θ)=−2/3 where 0≤θ≤π

2. Determine the quadrant of θ Since the cosine value is negative, θ must be in the second quadrant (Quadrant II), where the sine function is positive.

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

4. Substitute the known value of cos(θ) into the identity:

sin2(θ)+(−2/3)2=1

5. Solve for sin2(θ)

sin2(θ)+4/9=1

sin2(θ)=1−4/9

sin2(θ)=5/9

6. Take the square root and choose the positive sign because sin(θ)>0 in Quadrant II:

sin(θ)=√(,5/9)

sin(θ)=√(,5)/3

Final Answer

sin(arccos(−2/3))=√(,5)/3

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