(vi) Write down the Fourier series of the function $$ f(t) $$. Select the correct answer: $$ f(t)=9+18 \sin (t)-9 \sin (2 t)+6 \sin (3 t)+\ldots $$ $$ f(t)=9+36 \sin (t)-18 \sin (2 t)+6 \sin (3 t)+\ldots $$ $$ f(t)=9+18 \sin (t)-\frac{9}{2} \sin (2 t)+6 \sin (3 t)+\ldots $$ $$ f(t)=9+9 \sin (t)-18 \sin (2 t)+12 \sin (3 t)+\ldots $$ Correct response: $$ f(t)=9+18 \sin (t)-9 \sin (2 t)+6 \sin (3 t)+\ldots $$ (3 Mark) Click on Verify to continuel c) Determine the frequency of the Fourier series obtained in part (b)-(vi). Enter EXACT value in the form of say, 1/(3*PI) : Number (3 Mark)

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UpStudy AI · Accepted Answer

We first notice that the Fourier series is given as  
$$
f(t)=9+18\sin (t)-9\sin (2t)+6\sin (3t)+\ldots
$$

Each sine term is of the form $$\sin(n\,t)$$ where $$n=1, 2, 3, \ldots$$. Thus the argument of the sine functions is $$n\,t$$. In the standard Fourier series form for a function with period $$T$$, the sine terms have the argument
$$
\sin\left(n\,\omega_0\,t\right),
$$
where the fundamental angular frequency $$\omega_0$$ is given by
$$
\omega_0=\frac{2\pi}{T}.
$$

In our series the sine functions appear as $$\sin(n\,t)$$. This immediately tells us that  
$$
\omega_0=1\quad \mbox{(rad/s)}.
$$

Since the angular frequency $$\omega_0$$ is related to the frequency $$f$$ (in cycles per unit time) by
$$
\omega_0 = 2\pi\,f,
$$
we solve for $$f$$ as follows:
$$
f = \frac{\omega_0}{2\pi} = \frac{1}{2\pi}.
$$

Thus, the frequency of the Fourier series is  
$$
\frac{1}{2\pi}.
$$
The frequency of the Fourier series is $$ \frac{1}{2\pi} $$.

(vi) Write down the Fourier series of the function . Select the correct answer:

Correct response:

(3 Mark)
Click on Verify to continuel
c) Determine the frequency of the Fourier series obtained in part (b)-(vi). Enter EXACT value in the form of say,
1/(3*PI) :
Number
(3 Mark)

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(vi) Write down the Fourier series of the function . Select the correct answer: Correct response: (3 Mark) Click on Verify to continuel c) Determine the frequency of the Fourier series obtained in part (b)-(vi). Enter EXACT value in the form of say, 1/(3*PI) : Number (3 Mark)