Find the Exact Value sin((23pi)/6)

Problem

sin((23*π)/6)

Solution

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

(23*π)/6−2*π=(23*π)/6−(12*π)/6

(11*π)/6

2. Determine the quadrant for the angle (11*π)/6 Since (3*π)/2<(11*π)/6<2*π the angle is in Quadrant IV.

3. Find the reference angle (θ_R) for an angle in Quadrant IV using the formula (θ_R)=2*π−θ

(θ_R)=2*π−(11*π)/6

(θ_R)=π/6

4. Determine the sign of the sine function in Quadrant IV. In Quadrant IV, the ycoordinate is negative, so sin(θ) is negative.

sin((23*π)/6)=−sin(π/6)

5. Evaluate the sine of the reference angle using the unit circle or special triangles.

sin(π/6)=1/2

6. Apply the sign to the result.

−1/2

Final Answer

sin((23*π)/6)=−1/2

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