Simplify cos(theta)+cos(theta+2pi)+cos(theta+4pi)

Problem

cos(θ)+cos(θ+2*π)+cos(θ+4*π)

Solution

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

2. Simplify the second term using the periodicity identity where k=1

cos(θ+2*π)=cos(θ)

3. Simplify the third term using the periodicity identity where k=2

cos(θ+4*π)=cos(θ)

4. Substitute the simplified terms back into the original expression.

cos(θ)+cos(θ)+cos(θ)

5. Combine the like terms to find the final result.

3*cos(θ)

Final Answer

cos(θ)+cos(θ+2*π)+cos(θ+4*π)=3*cos(θ)

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