To evaluate \( f(x) = \sin x \) at \( x = \frac{13\pi}{2} \), we can first simplify \( \frac{13\pi}{2} \) by reducing it to an equivalent angle within the principal range of the sine function, which is \( [0, 2\pi) \).
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Find the equivalent angle by subtracting multiples of \( 2\pi \) from \( \frac{13\pi}{2} \): \[ 2\pi = \frac{4\pi}{1} \equiv \frac{8\pi}{2} \] So, subtract \( 2\pi \) twice to see where we land: \[ \frac{13\pi}{2} - 2(2\pi) = \frac{13\pi}{2} - \frac{8\pi}{2} = \frac{5\pi}{2} \]
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Now, we need to reduce \( \frac{5\pi}{2} \): \[ \frac{5\pi}{2} - 2\pi = \frac{5\pi}{2} - \frac{4\pi}{2} = \frac{\pi}{2} \]
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Now we can evaluate \( \sin \left( \frac{5\pi}{2} \right) \) which is equivalent to \( \sin \left( \frac{\pi}{2} \right) \): \[ \sin \left( \frac{\pi}{2} \right) = 1 \]
Thus, the final result is: \[ f\left( \frac{13\pi}{2} \right) = \sin \left( \frac{13\pi}{2} \right) = 1 \]
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