Find the Exact Value sin(6pi)

Problem

sin(6*π)

Solution

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

2. Rewrite the angle in terms of the period. Since 6*π is a multiple of 2*π we can express it as 3×2*π

3. Apply the periodicity property. Subtracting multiples of 2*π from the angle does not change the value of the sine function.

sin(6*π)=sin(6*π−3*(2*π))

4. Simplify the expression. The angle reduces to 0

sin(6*π)=sin(0)

5. Evaluate the sine of zero. On the unit circle, the ycoordinate at 0 radians is 0

sin(0)=0

Final Answer

sin(6*π)=0

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