Why does the video try to rewrite a periodic function as a sum of sines and cosines?

Conditions

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

Reasoning, step by step

1. Pose the central question of whether a periodic function can be represented as a sum of sines and cosines.
2. Explain that differential equations are often easy to solve with sines and cosines but less obvious for general functions like a square wave.
3. State that rewriting the square wave as a sum of sinusoids makes broader classes of problems tractable.
4. Mention that this decomposition is foundational to signal processing.

Example

The speaker asks whether a periodic function like this can be represented as an infinite sum of sines and cosines, and later notes that Fourier series were interesting in the study of differential equations because many are easy to solve with sines and cosines.

Common misconceptions

• Thinking that the decomposition is only a theoretical exercise without practical applications.
• Assuming that all differential equations are equally difficult to solve regardless of the input function's form.

Watch the explanation

Explore next

• What is the basic form of a Fourier series shown in this clip?
• Why are terms like $\cos (2t)$ and $\sin (3t)$ added when representing a square wave?