Simplify tan(arccos(x/2))

Problem

tan(arccos(x/2))

Solution

1. Define an angle θ such that θ=arccos(x/2) This implies cos(θ)=x/2 for 0≤θ≤π

2. Construct a right triangle where the adjacent side to angle θ is x and the hypotenuse is 2

3. Apply the Pythagorean theorem to find the length of the opposite side, b

x2+b2=2

b2=4−x2

b=√(,4−x2)

4. Determine the tangent of the angle θ using the ratio of the opposite side to the adjacent side.

tan(θ)=opposite/adjacent

tan(θ)=√(,4−x2)/x

Final Answer

tan(arccos(x/2))=√(,4−x2)/x

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