Find the Exact Value sin((16pi)/3)

Problem

sin((16*π)/3)

Solution

1. Find the coterminal angle by subtracting multiples of 2*π (which is (6*π)/3 until the angle is within the interval [0,2*π)

(16*π)/3−2*π=(10*π)/3

(10*π)/3−2*π=(4*π)/3

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

3. Find the reference angle θ′ for an angle in Quadrant III using the formula θ′=θ−π

θ′=(4*π)/3−π=π/3

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

sin((4*π)/3)=−sin(π/3)

5. Evaluate the sine of the reference angle using known trigonometric values.

sin(π/3)=√(,3)/2

6. Apply the sign to the result.

sin((16*π)/3)=−√(,3)/2

Final Answer

sin((16*π)/3)=−√(,3)/2

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