Simplify tan(arccos(10x))

Problem

tan(arccos(10*x))

Solution

1. Define an angle θ such that θ=arccos(10*x) This implies that cos(θ)=10*x where 0≤θ≤π

2. Represent the relationship using a right triangle where the adjacent side is 10*x and the hypotenuse is 1

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

a2+(10*x)2=1

a2=1−100*x2

a=√(,1−100*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)=√(,1−100*x2)/(10*x)

Final Answer

tan(arccos(10*x))=√(,1−100*x2)/(10*x)

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