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

Problem

sin(arctan(−√(,3)/3))

Solution

1. Evaluate the inner inverse trigonometric function arctan(−√(,3)/3)

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

3. Determine that tan(−π/6)=−1/√(,3)=−√(,3)/3 so θ=−π/6

4. Substitute the angle back into the outer function to get sin(−π/6)

5. Calculate the sine value using the unit circle or trigonometric properties.

6. Conclude that sin(−π/6)=−1/2

Final Answer

sin(arctan(−√(,3)/3))=−1/2

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