Evaluate sin((4pi)/3)

Problem

sin((4*π)/3)

Solution

1. Identify the quadrant of the angle (4*π)/3 Since π<(4*π)/3<(3*π)/2 the angle lies in Quadrant III.

2. Determine the reference angle. The reference angle θ′ for an angle in Quadrant III is calculated as θ′=θ−π

3. Calculate the reference angle: (4*π)/3−π=π/3

4. Apply the sine function to the reference angle. We know that sin(π/3)=√(,3)/2

5. Determine the sign of the sine function in Quadrant III. In Quadrant III, the sine value is negative.

6. Combine the sign and the reference value to find the final result.

Final Answer

sin((4*π)/3)=−√(,3)/2

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