Simplify arctan((- square root of 3)/3)

Problem

arctan(−√(,3)/3)

Solution

1. Identify the standard value for the tangent function that corresponds to the given ratio.

2. Recall the unit circle values where tan(θ)=√(,3)/3 This occurs when sin(θ)=1/2 and cos(θ)=√(,3)/2 which corresponds to θ=π/6

3. Apply the property of the arctangent function, which is defined on the interval (−π/2,π/2)

4. Use the odd function property of arctangent, which states arctan(−x)=−arctan(x)

5. Calculate the final value based on the negative ratio.

arctan(−√(,3)/3)=−arctan(√(,3)/3)

arctan(√(,3)/3)=π/6

arctan(−√(,3)/3)=−π/6

Final Answer

arctan(−√(,3)/3)=−π/6

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