Use a double-angle identity to find the exact value of each expression \( \sin \theta=-\frac{7}{25} \) and \( \frac{3 \pi}{2}<\theta<2 \pi \) Find \( \cos 2 \theta \)

To find \( \cos 2\theta \) given that \( \sin \theta = -\frac{7}{25} \) and \( \frac{3\pi}{2} < \theta < 2\pi \), we'll use the double-angle identity for cosine: \[ \cos 2\theta = 1 - 2\sin^2 \theta \] **Step 1: Compute \( \sin^2 \theta \)** \[ \sin^2 \theta = \left(-\frac{7}{25}\right)^2 = \frac{49}{625} \] **Step 2: Apply the Double-Angle Identity** \[ \cos 2\theta = 1 - 2 \times \frac{49}{625} = 1 - \frac{98}{625} = \frac{625}{625} - \frac{98}{625} = \frac{527}{625} \] Alternatively, you can use the identity: \[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta \] First, find \( \cos \theta \). Since \( \theta \) is in the fourth quadrant (\( \frac{3\pi}{2} < \theta < 2\pi \)), \( \cos \theta \) is positive. \[ \cos \theta = \sqrt{1 - \sin^2 \theta} = \sqrt{1 - \left(-\frac{7}{25}\right)^2} = \sqrt{1 - \frac{49}{625}} = \sqrt{\frac{576}{625}} = \frac{24}{25} \] Now, substitute back into the identity: \[ \cos 2\theta = \left(\frac{24}{25}\right)^2 - \left(-\frac{7}{25}\right)^2 = \frac{576}{625} - \frac{49}{625} = \frac{527}{625} \] **Final Answer:** \[ \cos 2\theta = \frac{527}{625} \]