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\( \mathbb{K} \) Use reference angles to find the exact value of the following expression. \[ \cos \frac{3 \pi}{4} \] Determine the reference angle for \( \frac{3 \pi}{4} \) The reference angle is

Ask by Collins Delgado. in the United States
Mar 10,2025

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The reference angle is \( \frac{\pi}{4} \).

Solución

1. Given the angle \( \theta = \frac{3\pi}{4} \), determine its quadrant. Since \[ \frac{\pi}{2} < \frac{3\pi}{4} < \pi, \] the angle lies in the second quadrant. 2. In the second quadrant, the reference angle \( \theta_{\text{ref}} \) is given by \[ \theta_{\text{ref}} = \pi - \theta. \] 3. Substitute the value of \( \theta \): \[ \theta_{\text{ref}} = \pi - \frac{3\pi}{4} = \frac{\pi}{4}. \] The reference angle is \( \frac{\pi}{4} \).

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The reference angle for \( \frac{3\pi}{4} \) is found by subtracting it from \( \pi \), which places it in the second quadrant. So, we calculate \( \pi - \frac{3\pi}{4} = \frac{\pi}{4} \). This means the reference angle is \( \frac{\pi}{4} \). Now, to find \( \cos \frac{3\pi}{4} \), we note that cosine is negative in the second quadrant. Thus, \( \cos \frac{3\pi}{4} = -\cos \frac{\pi}{4} = -\frac{\sqrt{2}}{2} \), giving us the exact value of \( \cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2} \).

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