If \( \sin \theta=\frac{12}{13}, 0<\theta<\frac{\pi}{2} \), find the exact value of each of the following. \( \begin{array}{lllll}\text { (a) } \sin (2 \theta) & \text { (b) } \cos (2 \theta) & \text { (c) } \sin \frac{\theta}{2} & \text { (d) } \cos \frac{\theta}{2} & \text { (e) } \tan 2 \theta\end{array} \) (f) \( \tan \frac{\theta}{2} \)

To solve the problem, we start with the given information: \( \sin \theta = \frac{12}{13} \). We can find \( \cos \theta \) using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \). 1. **Find \( \cos \theta \)**: \[ \cos^2 \theta = 1 - \sin^2 \theta = 1 - \left(\frac{12}{13}\right)^2 = 1 - \frac{144}{169} = \frac{25}{169} \] Therefore, \( \cos \theta = \sqrt{\frac{25}{169}} = \frac{5}{13} \) (since \( 0 < \theta < \frac{\pi}{2} \)). Now we can find the exact values for each part. (a) **Find \( \sin (2\theta) \)**: \[ \sin (2\theta) = 2 \sin \theta \cos \theta = 2 \cdot \frac{12}{13} \cdot \frac{5}{13} = \frac{120}{169} \] (b) **Find \( \cos (2\theta) \)**: Using the identity \( \cos (2\theta) = \cos^2 \theta - \sin^2 \theta \): \[ \cos (2\theta) = \left(\frac{5}{13}\right)^2 - \left(\frac{12}{13}\right)^2 = \frac{25}{169} - \frac{144}{169} = -\frac{119}{169} \] (c) **Find \( \sin \frac{\theta}{2} \)**: Using the half-angle formula \( \sin \frac{\theta}{2} = \sqrt{\frac{1 - \cos \theta}{2}} \): \[ \sin \frac{\theta}{2} = \sqrt{\frac{1 - \frac{5}{13}}{2}} = \sqrt{\frac{\frac{8}{13}}{2}} = \sqrt{\frac{8}{26}} = \sqrt{\frac{4}{13}} = \frac{2}{\sqrt{13}} = \frac{2\sqrt{13}}{13} \] (d) **Find \( \cos \frac{\theta}{2} \)**: Using the half-angle formula \( \cos \frac{\theta}{2} = \sqrt{\frac{1 + \cos \theta}{2}} \): \[ \cos \frac{\theta}{2} = \sqrt{\frac{1 + \frac{5}{13}}{2}} = \sqrt{\frac{\frac{18}{13}}{2}} = \sqrt{\frac{18}{26}} = \sqrt{\frac{9}{13}} = \frac{3}{\sqrt{13}} = \frac{3\sqrt{13}}{13} \] (e) **Find \( \tan (2\theta) \)**: Using the identity \( \tan (2\theta) = \frac{\sin (2\theta)}{\cos (2\theta)} \): \[ \tan (2\theta) = \frac{\frac{120}{169}}{-\frac{119}{169}} = -\frac{120}{119} \] (f) **Find \( \tan \frac{\theta}{2} \)**: Using the identity \( \tan \frac{\theta}{2} = \frac{\sin \theta}{1 + \cos \theta} \): \[ \tan \frac{\theta}{2} = \frac{\frac{12}{13}}{1 + \frac{5}{13}} = \frac{\frac{12}{13}}{\frac{18}{13}} = \frac{12}{18} = \frac{2}{3} \] So, the final answers are: - (a) \( \sin (2\theta) = \frac{120}{169} \) - (b) \( \cos (2\theta) = -\frac{119}{169} \) - (c) \( \sin \frac{\theta}{2} = \frac{2\sqrt{13}}{13} \) - (d) \( \cos \frac{\theta}{2} = \frac{3\sqrt{13}}{13} \) - (e) \( \tan (2\theta) = -\frac{120}{119} \) - (f) \( \tan \frac{\theta}{2} = \frac{2}{3} \)