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If the base period is $2\pi$, the term $\cos(t)$ has a frequency of $\frac{1}{2\pi}$ cycles per second. Doubling the argument to $\cos(2t)$ doubles the frequency, so $\cos(2t)$ has a frequency of $\frac{1}{\pi}$ cycles per second.

Conditions: The base period of the function is $2\pi$.; The frequency is defined as the reciprocal of the period.

If the base period is $2\pi$, the term $\cos(t)$ has a frequency of $\frac{1}{2\pi}$ cycles per second. Doubling the argument to $\cos(2t)$ doubles the frequency, so $\cos(2t)$ has a frequency of $\frac{1}{\pi}$ cycles per second.

Conditions: The base period of the function is $2\pi$.; The frequency is defined as the reciprocal of the period.

Terms like $\cos(2t)$ and $\sin(3t)$ are added because a single fundamental sine-cosine pair ($a_1\cos(t)+b_1\sin(t)$) only produces a smooth sinusoidal shape. To match the sharp transitions of a square wave, the series must include higher harmonics whose frequencies are integer multiples of the base frequency.

Conditions: The target function is a square wave with period $2\pi$.; The representation uses sine and cosine terms.

Terms like $\cos(2t)$ and $\sin(3t)$ are added because a single fundamental sine-cosine pair ($a_1\cos(t)+b_1\sin(t)$) only produces a smooth sinusoidal shape. To match the sharp transitions of a square wave, the series must include higher harmonics whose frequencies are integer multiples of the base frequency.

Conditions: The target function is a square wave with period $2\pi$.; The representation uses sine and cosine terms.