Find the Exact Value cos(11pi)

Problem

cos(11*π)

Solution

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

2. Reduce the angle by subtracting multiples of the period. Since 11*π is an odd multiple of π we can write it as 10*π+π

3. Apply the periodicity rule. Because 10*π is 5 full rotations (5×2*π, the value of the function at 11*π is the same as the value at π

cos(11*π)=cos(π+5*(2*π))

cos(11*π)=cos(π)

4. Evaluate the cosine of the resulting angle using the unit circle. At the angle π (or 180, the coordinates on the unit circle are (−1,0) Since cos(θ) corresponds to the xcoordinate:

cos(π)=−1

Final Answer

cos(11*π)=−1

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