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

Problem

tan(arccos(−2/3))

Solution

1. Identify the inner function and let θ=arccos(−2/3) This implies cos(θ)=−2/3

2. Determine the range of the inverse cosine function. Since the range of arccos(x) is [0,π] and the cosine value is negative, θ must be in Quadrant II.

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

sin2(θ)+(−2/3)2=1

sin2(θ)+4/9=1

sin2(θ)=5/9

4. Solve for sin(θ) Since θ is in Quadrant II, the sine value must be positive.

sin(θ)=√(,5/9)

sin(θ)=√(,5)/3

5. Apply the definition of the tangent function, tan(θ)=sin(θ)/cos(θ) to find the final value.

tan(θ)=√(,5)/3/(−2/3)

tan(θ)=−√(,5)/2

Final Answer

tan(arccos(−2/3))=−√(,5)/2

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