Why are terms like $\cos (2t)$ and $\sin (3t)$ added when representing a square wave?

Conditions

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

Reasoning, step by step

1. Start with the fundamental terms $a_1\cos(t)$ and $b_1\sin(t)$.
2. Observe that this combination alone looks like a very clean sinusoid, not a square wave.
3. Recognize the need for additional components to capture the sharp corners.
4. Add sinusoids whose frequencies are multiples of the base frequency, such as $\cos(2t)$ and $\sin(3t)$.

Example

The speaker says that $a_1\cos(t)+b_1\sin(t)$ by itself is not going to describe the function because it would look like a very clean sinusoid, not like a square wave, so they add sinusoids of frequencies that are multiples of these frequencies.

Common misconceptions

• Mistaking a single fundamental sinusoid for a full square-wave representation.
• Overlooking the effect of the coefficient inside the trigonometric argument, not realizing that $\cos(2t)$ has twice the frequency of $\cos(t)$.

Watch the explanation

Explore next

• If the base period is 2π, what frequency does $\cos (2t)$ have?
• What do the coefficients a₁ and b₁ mean in the series?