Find the Value Using the Unit Circle sin((4pi)/3)

Problem

sin((4*π)/3)

Solution

1. Identify the location of the angle (4*π)/3 on the unit circle. Since π<(4*π)/3<(3*π)/2 the angle is in the third quadrant.

2. Determine the reference angle by subtracting π from the given angle.

(θ_ref)=(4*π)/3−π=π/3

3. Recall the sine value for the reference angle π/3 from the unit circle.

sin(π/3)=√(,3)/2

4. Apply the sign based on the quadrant. In the third quadrant, the ycoordinates (sine values) are negative.

sin((4*π)/3)=−√(,3)/2

Final Answer

sin((4*π)/3)=−√(,3)/2

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