What is the period of $f (t) = sin (\frac{t}{2}) + cos (\frac{7 t}{24})$ $f(t)=sin( t /2 )+ cos( (7t)/24 )$ ?

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Answer 1 · 2018-04-21T06:41:33

The period is $= 48 π$ $=48pi$

A periodic function is such that

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

where $T$ $T$ is the period

Therefore,

$sin (\frac{7}{24} t) = sin (\frac{7}{24} (t + T))$ $sin(7/24t)=sin(7/24(t+T))$

$= sin (\frac{7}{24} t + \frac{7}{24} T)$ $=sin(7/24t+7/24T)$

$= sin (\frac{7}{24} t) cos (\frac{7}{24} T) + sin (\frac{7}{24} T) cos (\frac{7}{24} t)$ $=sin(7/24t)cos(7/24T)+sin(7/24T)cos(7/24t)$

Comparing the $L H S$ $LHS$ and the $R H S$ $RHS$

${(cos(7/24T)=1),(sin(7/24T)cos(7/24t)=0):}$

$<=>$ , ${(7/24T=14pi):}$

$T=48pi$

$sin(t/2)=sin(1/2(t+T))$

$=sin(1/2t+1/2T)$

$=sin(t/2)cos(T/2)+sin(T/2)cos(t/2)$

Comparing the $LHS$ and the $RHS$

${(cos(T/2)=1),(sin(T/2)cos(t/2)=0):}$

$<=>$ , ${(T/2=2pi),(T/2=0):}$

$T=4pi$

The $LCM$ of $4pi "and" 48pi$ is $=48pi$

What is the period of f(t)=sin( t /2 )+ cos( (7t)/24 ) f(t)=sin(t2)+cos(7t24)f(t)=sin( t /2 )+ cos( (7t)/24 ) ?