# What is the frequency of f(t)= sin(4t) - cos(7t)?

Apr 23, 2017

${f}_{0} = \frac{1}{2 \pi} \text{ Hz}$

#### Explanation:

Given: $f \left(t\right) = \sin \left(4 t\right) - \cos \left(7 t\right)$ where t is seconds.

Use this reference for Fundamental Frequency

Let ${f}_{0}$ be the fundamental frequency of the combined sinusoids, in Hz (or ${\text{s}}^{-} 1$).

${\omega}_{1} = 4 \text{ rad/s}$
${\omega}_{2} = 7 \text{ rad/s}$

Using the fact that $\omega = 2 \pi f$

${f}_{1} = \frac{4}{2 \pi} = \frac{2}{\pi} \text{ Hz}$ and ${f}_{2} = \frac{7}{2 \pi} \text{ Hz}$

The fundamental frequency is the greatest common divisor of the two frequencies:

${f}_{0} = \gcd \left(\frac{2}{\pi} \text{ Hz", 7/(2pi)" Hz}\right)$

${f}_{0} = \frac{1}{2 \pi} \text{ Hz}$

Here is a graph:

graph{y = sin(4x) - cos(7x) [-10, 10, -5, 5]}

Please observe that it repeats every $2 \pi$