What is the frequency of $f (θ) = sin 24 t - cos 42 t$ $f(theta)= sin 24 t - cos 42 t$ ?

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Answer 1 · 2018-07-28T13:46:35

The frequency is $f = \frac{3}{π}$ $f=3/pi$

The period $T$ $T$ of a periodic function $f (x)$ $f(x)$ is given by

$f (x) = f (x + T)$ $f(x)=f(x+T)$

Here,

$f (t) = sin 24 t - cos 42 t$ $f(t)=sin24t-cos42t$

Therefore,

$f (t + T) = sin 24 (t + T) - cos 42 (t + T)$ $f(t+T)=sin24(t+T)-cos42(t+T)$

$= sin (24 t + 24 T) - cos (42 t + 42 T)$ $=sin(24t+24T)-cos(42t+42T)$

$= sin 24 t cos 24 T + cos 24 t sin 24 T - cos 42 t cos 42 T + sin 42 t sin 42 T$ $=sin24tcos24T+cos24tsin24T-cos42tcos42T+sin42tsin42T$

Comparing,

$f (t) = f (t + T)$ $f(t)=f(t+T)$

${(cos24T=1),(sin24T=0),(cos42T=1),(sin42T=0):}$

$<=>$ , ${(24T=2pi),(42T=2pi):}$

$<=>$ , ${(T=1/12pi=7/84pi),(T=4/84pi):}$

The LCM of $7/84pi$ and $4/84pi$ is

$=28/84pi=1/3pi$

The period is $T=1/3pi$

The frequency is

$f=1/T=1/(1/3pi)=3/pi$

graph{sin(24x)-cos(42x) [-1.218, 2.199, -0.82, 0.889]}

What is the frequency of f(theta)= sin 24 t - cos 42 t f(θ)=sin24t−cos42tf(theta)= sin 24 t - cos 42 t ?