Simplify tan(arccos(3x))

Problem

tan(arccos(3*x))

Solution

1. Identify the inner function and set it equal to an angle θ Let θ=arccos(3*x)

2. Rewrite the inverse trigonometric relationship as a trigonometric equation. This gives cos(θ)=3*x where 0≤θ≤π

3. Represent the relationship using a right triangle. Since cos(θ)=adjacent/hypotenuse we can let the adjacent side be 3*x and the hypotenuse be 1

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

(3*x)2+opposite2=1

opposite2=1−9*x2

opposite=√(,1−9*x2)

5. Evaluate the original expression tan(θ) using the sides of the triangle. Since tan(θ)=opposite/adjacent we substitute the known values.

tan(θ)=√(,1−9*x2)/(3*x)

Final Answer

tan(arccos(3*x))=√(,1−9*x2)/(3*x)

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