Find the Exact Value tan(arccos(3/4))

Problem

tan(arccos(3/4))

Solution

1. Identify the inner expression as an angle θ such that θ=arccos(3/4)

2. Rewrite the expression using the definition of the inverse cosine function, which implies cos(θ)=3/4 for 0≤θ≤π

3. Relate the trigonometric functions using a right triangle where the adjacent side is 3 and the hypotenuse is 4

4. Calculate the opposite side y using the Pythagorean theorem 3^2+y^2=4^2

5. Solve for y to find y=√(,16−9)=√(,7)

6. Determine the value of tan(θ) using the ratio of the opposite side to the adjacent side, which is y/3

Final Answer

tan(arccos(3/4))=√(,7)/3

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