(v) Determine the angular frequency of the fundamental ( 1 st ) harmonic, $$ 2 n \mathrm{nd}, 3 \mathrm{rd}, 4 $$ th and the 5 th harmonics, Note: Enter your answer in the form of say $$ (1,2 / 3,3 / 2,3,5) $$. The numbers are exact values and enclosed in brackets and separated by a comma. Take note of the * sign. $$ \left(\omega_{1}, \omega_{2}, \omega_{3}, \omega_{4}, \omega_{5}\right)=(1,2,3,4,5) $$ correct responsel $$ (1 \quad 2 \quad 3 \quad 45) $$ (3 Mark) Click on Verify to continuel (vi) Write down the Fourier $$ \operatorname{series} $$ of the function $$ f(t) $$. Select the correct answer: O $$ f(t)=9+18 \sin (t)-9 \sin (2 t)+6 \sin (3 t)+\ldots $$ O $$ (t)=9+36 \sin (t)-18 \sin (2 t)+6 \sin (3 t)+\ldots $$ O $$ (t)=9+18 \sin (t)-\frac{9}{2} \sin (2 t)+6 \sin (3 t)+\ldots $$ ( 3 Mark) Click on Verify to $$ \operatorname{continuel~} $$ Section Attempt 1 of 1 Verify

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For the fundamental harmonic and the first five harmonics, the angular frequencies can be expressed as $$ \omega_n = n \cdot \omega_1 $$, where $$ n $$ is the harmonic number. Given that $$ \omega_1 = 1 $$, we can find the angular frequencies for the first five harmonics.
Thus, we have:
- $$ \omega_1 = 1 $$
- $$ \omega_2 = 2 $$
- $$ \omega_3 = 3 $$
- $$ \omega_4 = 4 $$
- $$ \omega_5 = 5 $$

The angular frequencies are $$ (1, 2, 3, 4, 5) $$.Final Answer for Angular FrequenciesThe angular frequencies of the fundamental and the first five harmonics are $$ (1, 2, 3, 4, 5) $$.
Now, let's analyze the given options for the Fourier series of the function $$ f(t) $$. The Fourier series typically consists of a constant term and sine terms with coefficients that depend on the function being represented.
We need to identify the correct Fourier series from the provided options:
1. $$ f(t) = 9 + 18 \sin(t) - 9 \sin(2t) + 6 \sin(3t) + \ldots $$
2. $$ f(t) = 9 + 36 \sin(t) - 18 \sin(2t) + 6 \sin(3t) + \ldots $$
3. $$ f(t) = 9 + 18 \sin(t) - \frac{9}{2} \sin(2t) + 6 \sin(3t) + \ldots $$

To determine the correct series, we need to check the coefficients of the sine terms. The first option has a coefficient of $$ -9 $$ for $$ \sin(2t) $$, which seems reasonable. The second option has a coefficient of $$ -18 $$, which is too high. The third option has $$ -\frac{9}{2} $$, which is also not consistent with the expected pattern.
The correct Fourier series of the function $$ f(t) $$ is $$ f(t) = 9 + 18 \sin(t) - 9 \sin(2t) + 6 \sin(3t) + \ldots $$.

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