Find the Exact Value sin((10pi)/3)

Problem

sin((10*π)/3)

Solution

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

(10*π)/3−2*π=(10*π)/3−(6*π)/3

(10*π)/3−(6*π)/3=(4*π)/3

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

3. Identify the sign of the sine function in Quadrant III. In this quadrant, the ycoordinate is negative, so sin((4*π)/3) is negative.

4. Calculate the reference angle θ^′ for (4*π)/3 using the formula θ^′=θ−π

θ^′=(4*π)/3−π

θ^′=π/3

5. Evaluate the sine of the reference angle and apply the negative sign.

sin(π/3)=√(,3)/2

sin((4*π)/3)=−√(,3)/2

Final Answer

sin((10*π)/3)=−√(,3)/2

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