Simplify 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. Apply the property of odd functions for arctangent, which states arctan(−x)=−arctan(x)

3. Rewrite the expression using this property:

arctan(−√(,3)/3)=−arctan(√(,3)/3)

4. Rationalize or recognize the value √(,3)/3 as 1/√(,3)

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

6. Recall from the unit circle that tan(π/6)=1/√(,3)

7. Substitute the value back into the expression:

−arctan(√(,3)/3)=−π/6

Final Answer

arctan(−√(,3)/3)=−π/6

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