Find the Exact Value sin((19pi)/6)

Problem

sin((19*π)/6)

Solution

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

2*π=(12*π)/6

(19*π)/6−(12*π)/6=(7*π)/6

2. Determine the quadrant of the angle (7*π)/6 Since π<(7*π)/6<(3*π)/2 the angle lies in Quadrant III.

3. Find the reference angle for (7*π)/6 in Quadrant III.

(θ_ref)=(7*π)/6−π

(θ_ref)=π/6

4. Determine the sign of the sine function in Quadrant III. In Quadrant III, the ycoordinates are negative, so sin(θ)<0

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

sin(π/6)=1/2

sin((19*π)/6)=−1/2

Final Answer

sin((19*π)/6)=−1/2

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