Find the Exact Value sin((29pi)/6)

Problem

sin((29*π)/6)

Solution

1. Find the coterminal angle by subtracting multiples of 2*π (which is (12*π)/6 until the angle is within the interval [0,2*π)

(29*π)/6−2*((12*π)/6)=(29*π)/6−(24*π)/6

(29*π)/6≡(5*π)/6

2. Identify the quadrant for the angle (5*π)/6 Since π/2<(5*π)/6<π the angle is in Quadrant II.

3. Determine the reference angle for (5*π)/6 in Quadrant II.

π−(5*π)/6=π/6

4. Apply the sine sign rule for Quadrant II. Since sine is positive in Quadrant II, the value is equal to the sine of the reference angle.

sin((5*π)/6)=sin(π/6)

5. Evaluate using the known value for sin(π/6)

sin(π/6)=1/2

Final Answer

sin((29*π)/6)=1/2

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