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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.

The video aims to represent a complex periodic function, like a square wave, as an infinite sum of simpler sinusoidal components. This decomposition is highly useful because many differential equations are much easier to solve when the inputs are sines and cosines, and it serves as the foundation for signal processing.

Conditions: The target function is periodic.; The proposed representation uses sines and cosines with different periods or frequencies.

The video aims to represent a complex periodic function, like a square wave, as an infinite sum of simpler sinusoidal components. This decomposition is highly useful because many differential equations are much easier to solve when the inputs are sines and cosines, and it serves as the foundation for signal processing.

Conditions: The target function is periodic.; The proposed representation uses sines and cosines with different periods or frequencies.