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

Problem

sin(arctan(−√(,3)))

Solution

1. Evaluate the inner expression arctan(−√(,3))

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

3. Determine that θ=−π/3 because tan(−π/3)=−√(,3)

4. Substitute the angle back into the sine function to get sin(−π/3)

5. Calculate the value of sin(−π/3) using the unit circle or trigonometric properties.

6. Apply the property sin(−x)=−sin(x) which gives −sin(π/3)=−√(,3)/2

Final Answer

sin(arctan(−√(,3)))=−√(,3)/2

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