To complete the table of values for the function \( f(x) = \frac{7}{3} \sin x \) at the specified x-values, we will calculate \( f(x) \) at each given value of \( x \):
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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 we can summarize the completed table of values:
| \( x \) | \( f(x) = \frac{7}{3} \sin x \) | |------------------|----------------------------------| | \( 0 \) | \( 0 \) | | \( \frac{\pi}{2} \) | \( \frac{7}{3} \) | | \( \pi \) | \( 0 \) | | \( \frac{3\pi}{2} \) | \( -\frac{7}{3} \) | | \( 2\pi \) | \( 0 \) |
This table provides the necessary values to graph the function \( f(x) = \frac{7}{3} \sin x \) over one complete period.