To find the values of the function \( f(x) = \cos(6x) \) at the specified points \( x = 0, \frac{\pi}{12}, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3} \), we will calculate the cosine values step by step.
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For \( x = 0 \): \[ f(0) = \cos(6 \cdot 0) = \cos(0) = 1 \]
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For \( x = \frac{\pi}{12} \): \[ f\left(\frac{\pi}{12}\right) = \cos\left(6 \cdot \frac{\pi}{12}\right) = \cos\left(\frac{6\pi}{12}\right) = \cos\left(\frac{\pi}{2}\right) = 0 \]
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For \( x = \frac{\pi}{6} \): \[ f\left(\frac{\pi}{6}\right) = \cos\left(6 \cdot \frac{\pi}{6}\right) = \cos(6 \cdot \frac{\pi}{6}) = \cos(\pi) = -1 \]
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For \( x = \frac{\pi}{4} \): \[ f\left(\frac{\pi}{4}\right) = \cos\left(6 \cdot \frac{\pi}{4}\right) = \cos\left(\frac{6\pi}{4}\right) = \cos\left(\frac{3\pi}{2}\right) = 0 \]
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For \( x = \frac{\pi}{3} \): \[ f\left(\frac{\pi}{3}\right) = \cos\left(6 \cdot \frac{\pi}{3}\right) = \cos(2\pi) = 1 \]
Now we can summarize the results:
- \( f(0) = 1 \)
- \( f\left(\frac{\pi}{12}\right) = 0 \)
- \( f\left(\frac{\pi}{6}\right) = -1 \)
- \( f\left(\frac{\pi}{4}\right) = 0 \)
- \( f\left(\frac{\pi}{3}\right) = 1 \)
Thus, the values are:
- \( f(0) = 1 \)
- \( f\left(\frac{\pi}{12}\right) = 0 \)
- \( f\left(\frac{\pi}{6}\right) = -1 \)
- \( f\left(\frac{\pi}{4}\right) = 0 \)
- \( f\left(\frac{\pi}{3}\right) = 1 \)