Simplify tan(arccos(x/3))

Problem

tan(arccos(x/3))

Solution

1. Identify the inner expression as an angle θ where θ=arccos(x/3)

2. Apply the definition of the inverse cosine function, which implies cos(θ)=x/3 for 0≤θ≤π

3. Construct a right triangle where the adjacent side is x and the hypotenuse is 3

4. Determine the opposite side using the Pythagorean theorem a2+b2=c2

x2+opposite2=3

opposite2=9−x2

opposite=√(,9−x2)

5. Evaluate the tangent of the angle θ using the ratio of the opposite side to the adjacent side.

tan(θ)=opposite/adjacent

tan(θ)=√(,9−x2)/x

Final Answer

tan(arccos(x/3))=√(,9−x2)/x

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