# What is the frequency of f(theta)= sin 6 t - cos 21 t ?

Aug 12, 2016

$\frac{3}{2 \pi} = 0.4775$, nearly.

#### Explanation:

The period for both sin kt and cos kt is $2 \frac{\pi}{k}$.

The periods for the separate oscillations $\sin 6 t \mathmr{and} - \cos 21 t$ are

$\frac{\pi}{3} \mathmr{and} \frac{2 \pi}{21}$, respectively.

Twice the first is seven times the second. This common value

(least)  P = (2pi)/3) is the period for the compounded oscillation f(t).

See how it works.

$f \left(t + P\right)$

$= f \left(t + \frac{2 \pi}{3}\right)$

=sin((6t+4pi)-cos(21t+14pi)

$= \sin 6 t - \cos 21 t$

=f(t).

Note that P/2 used instead of P changes the sign of the second

term. .

Frequency is 1/P..