Find the Value Using the Unit Circle sin((5pi)/3)

Problem

sin((5*π)/3)

Solution

1. Identify the position of the angle (5*π)/3 on the unit circle. Since (5*π)/3 is between (3*π)/2 and 2*π the terminal side of the angle lies in Quadrant IV.

2. Determine the reference angle θ′ In Quadrant IV, the reference angle is calculated by subtracting the angle from 2*π

θ′=2*π−(5*π)/3

θ′=π/3

3. Evaluate the sine of the reference angle. Using the known values for special angles on the unit circle:

sin(π/3)=√(,3)/2

4. Apply the correct sign based on the quadrant. In Quadrant IV, the ycoordinate (sine) is negative.

sin((5*π)/3)=−sin(π/3)

sin((5*π)/3)=−√(,3)/2

Final Answer

sin((5*π)/3)=−√(,3)/2

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