To find the values of the function \( f(x) = \frac{7}{3} \sin(x) \) at the given points, we can evaluate \( f(x) \) at each of the specified values of \( x \): \( 0 \), \( \frac{\pi}{2} \), \( \pi \), \( \frac{3\pi}{2} \), and \( 2\pi \).
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For \( x = 0 \): \[ f(0) = \frac{7}{3} \sin(0) = \frac{7}{3} \cdot 0 = 0 \]
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For \( x = \frac{\pi}{2} \): \[ f\left(\frac{\pi}{2}\right) = \frac{7}{3} \sin\left(\frac{\pi}{2}\right) = \frac{7}{3} \cdot 1 = \frac{7}{3} \]
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For \( x = \pi \): \[ f(\pi) = \frac{7}{3} \sin(\pi) = \frac{7}{3} \cdot 0 = 0 \]
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For \( x = \frac{3\pi}{2} \): \[ f\left(\frac{3\pi}{2}\right) = \frac{7}{3} \sin\left(\frac{3\pi}{2}\right) = \frac{7}{3} \cdot (-1) = -\frac{7}{3} \]
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For \( x = 2\pi \): \[ f(2\pi) = \frac{7}{3} \sin(2\pi) = \frac{7}{3} \cdot 0 = 0 \]
Now, summarizing the values:
- \( f(0) = 0 \)
- \( f\left(\frac{\pi}{2}\right) = \frac{7}{3} \)
- \( f(\pi) = 0 \)
- \( f\left(\frac{3\pi}{2}\right) = -\frac{7}{3} \)
- \( f(2\pi) = 0 \)
Thus, the values are:
\[ f(x) = \begin{cases} 0 & \text{for } x = 0, \pi, 2\pi \ \frac{7}{3} & \text{for } x = \frac{\pi}{2} \ -\frac{7}{3} & \text{for } x = \frac{3\pi}{2} \end{cases} \]