Find the Exact Value arctan(-( square root of 3)/3)

Problem

arctan(−√(,3)/3)

Solution

1. Identify the range of the inverse tangent function, which is (−π/2,π/2)

2. Rewrite the expression inside the function by rationalizing or simplifying to recognize a standard trigonometric ratio.

3. Observe that √(,3)/3 is equivalent to 1/√(,3)

4. Recall the tangent values for standard angles where tan(θ)=sin(θ)/cos(θ)

5. Determine the angle θ such that tan(θ)=−1/√(,3) within the interval (−π/2,π/2)

6. Conclude that since tan(π/6)=1/√(,3) then tan(−π/6)=−1/√(,3) because tangent is an odd function.

Final Answer

arctan(−√(,3)/3)=−π/6

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