Simplify 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 is in the third quadrant.

2. Determine the sign of the sine function in the third quadrant. In the third quadrant, the sine of an angle is negative.

3. Find the reference angle. The reference angle θ′ is calculated by subtracting π from the angle.

θ′=(4*π)/3−π

θ′=π/3

4. Apply the reference angle property. The value of sin((4*π)/3) is equal to −sin(π/3)

5. Evaluate the sine of the reference angle. Using the unit circle or special triangles, sin(π/3)=√(,3)/2

sin((4*π)/3)=−√(,3)/2

Final Answer

sin((4*π)/3)=−√(,3)/2

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