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

Problem

sin(arctan(−√(,3)/3))

Solution

1. Identify the inner value θ=arctan(−√(,3)/3) By the definition of the inverse tangent function, θ must lie in the interval (−π/2,π/2)

2. Determine the angle θ such that tan(θ)=−√(,3)/3 Since tan(π/6)=1/√(,3)=√(,3)/3 and the tangent function is odd, it follows that θ=−π/6

3. Substitute the value of θ back into the original expression to find sin(θ)

4. Evaluate the sine of the angle: sin(−π/6) Using the unit circle or the property that sine is an odd function, sin(−π/6)=−sin(π/6)=−1/2

Final Answer

sin(arctan(−√(,3)/3))=−1/2

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