Simplify tan(arccos(-( square root of 3)/2))

Problem

tan(arccos(−√(,3)/2))

Solution

1. Identify the inner value of the inverse cosine function, which is −√(,3)/2

2. Determine the angle θ=arccos(−√(,3)/2) within the restricted range of the arccosine function, which is [0,π]

3. Recall the unit circle values where cos(θ)=−√(,3)/2 in the second quadrant.

4. Evaluate the angle to be θ=(5*π)/6

5. Substitute the angle back into the tangent function to get tan((5*π)/6)

6. Calculate the tangent value using the ratio sin(θ)/cos(θ)

7. Simplify the expression (1/2)/(−√(,3)/2) to find the final value.

Final Answer

tan(arccos(−√(,3)/2))=−√(,3)/3

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