if x(t)=sin10*pi*t calculate trigonometric fourier coefficients and T=1/3

Given the function \( x(t) = \sin(10\pi t) \) and the period \( T = \frac{1}{3} \), we are to calculate the trigonometric Fourier coefficients.

### 1. Determining the Fundamental Frequency
The fundamental frequency \( f_0 \) is given by:
\[
f_0 = \frac{1}{T} = 3 \, \text{Hz}
\]
The angular frequency \( \omega_0 \) is:
\[
\omega_0 = 2\pi f_0 = 6\pi \, \text{rad/s}
\]

### 2. Express \( x(t) \) in terms of \( \omega_0 \)
The given function \( x(t) = \sin(10\pi t) \) can be rewritten as:
\[
x(t) = \sin\left(\frac{10\pi}{6\pi} \cdot 6\pi t\right) = \sin\left(\frac{5}{3} \cdot 6\pi t\right)
\]
This shows that \( x(t) = \sin\left(5 \omega_0 t\right) \).

### 3. Trigonometric Fourier Series
The Fourier series for \( x(t) \) is given by:
\[
x(t) = \sum_{n=-\infty}^{\infty} C_n e^{jn\omega_0 t}
\]
However, since \( x(t) \) is purely sinusoidal, it will have non-zero coefficients only for \( n = \pm 5 \).

### 4. Fourier Coefficients
For \( n = 5 \):
\[
C_5 = \frac{1}{2j}
\]
For \( n = -5 \):
\[
C_{-5} = -\frac{1}{2j}
\]

All other Fourier coefficients \( C_n \) for \( n \neq \pm 5 \) are zero.

### Final Answer
The trigonometric Fourier coefficients for \( x(t) = \sin(10\pi t) \) with \( T = \frac{1}{3} \) are:
\[
C_5 = \frac{1}{2j}, \quad C_{-5} = -\frac{1}{2j}, \quad \text{and } C_n = 0 \text{ for all } n \neq \pm 5
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you compute the Fourier series for a more complex periodic function?
2. What happens to the Fourier coefficients if the period \( T \) is changed?
3. Can the Fourier coefficients be computed directly from the integral formulas?
4. How would the Fourier series differ if \( x(t) \) were a cosine function instead of sine?
5. What is the significance of the magnitude and phase of Fourier coefficients?

**Tip:** Understanding the symmetry of the function can simplify the calculation of Fourier coefficients. For example, odd functions only have sine terms in their Fourier series.

Calculate the trigonometric Fourier coefficients for x(t) = sin(10πt) with a period of T = 1/3. This problem covers key concepts in Fourier analysis, including fundamental frequency, angular frequency, and the trigonometric Fourier series. Suitable for undergraduate students studying signal processing or applied mathematics.

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