Why does a larger $a_1$ than $a_2$ mean more of the $1/(2π)$ hertz frequency?

Conditions

• The function is represented in the displayed form with cosine coefficients $a_1, a_2, a_3$.
• The comparison is about the coefficients multiplying $\cos(t)$ and $\cos(2t)$.

Reasoning, step by step

1. Identify the frequency associated with $\cos(t)$ as $\frac{1}{2\pi}$.
2. Identify the frequency associated with $\cos(2t)$ as $\frac{1}{\pi}$.
3. Interpret the coefficients $a_1$ and $a_2$ as the amounts of these respective frequencies present in the function.
4. Conclude that $a_1 \gg a_2$ implies more of the $\frac{1}{2\pi}$ hertz frequency than the $\frac{1}{\pi}$ frequency.

Example

The speaker says, 'if a1 is much bigger than a2, then that tells you that that function contains a lot more of the 1 over 2 pi hertz frequency than the 1 over pi frequency.'

Common misconceptions

• Thinking the coefficients are just weights on cosines and sines without realizing they also tell how much of different frequencies the function contains.
• Confusing the time-domain view of $f(t)$ with the frequency-domain view.

Watch the explanation

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