Find the Exact Value sin(12pi)

Problem

sin(12*π)

Solution

1. Identify the period of the sine function. The sine function sin(θ) has a period of 2*π meaning sin(θ+2*π*k)=sin(θ) for any integer k

2. Determine the number of full rotations. Since 12*π is an even multiple of π we can write it as 6×2*π

3. Apply the periodicity property. Because 12*π is a multiple of 2*π the angle is coterminal with 0

sin(12*π)=sin(0+6×2*π)

sin(12*π)=sin(0)

4. Evaluate the sine of the coterminal angle. On the unit circle, the ycoordinate at 0 radians is 0

sin(0)=0

Final Answer

sin(12*π)=0

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