Find the Exact Value cos(12pi)

Problem

cos(12*π)

Solution

1. Identify the period of the cosine function. The cosine function cos(θ) has a period of 2*π meaning cos(θ)=cos(θ+2*π*k) 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 a multiple of the period 2*π

12*π=6*(2*π)

3. Simplify the angle using periodicity. An angle of 12*π is coterminal with 0 on the unit circle.

cos(12*π)=cos(0)

4. Evaluate the cosine of the resulting angle. On the unit circle, the xcoordinate at 0 radians is 1

cos(0)=1

Final Answer

cos(12*π)=1

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