1. Find the exact value of each expression, using the appropriate double-angle or half-angle formula. a) If $\sin \theta=-\frac{5}{13}, \theta$ in quadrant III, find $\sin 2 \theta$. b) If $\cot \theta=-\frac{3}{4}, \theta$ in quadrant IV, find $\cos \frac{\theta}{2}$

Step 1 ：Given that \(\sin \theta = -\frac{5}{13}\) and \(\theta\) is in quadrant III, we need to find \(\sin 2 \theta\).

Step 2 ：We can use the double angle formula for sine, which is \(\sin 2\theta = 2\sin\theta\cos\theta\). We know the value of \(\sin\theta\), but we need to find the value of \(\cos\theta\).

Step 3 ：Since \(\theta\) is in quadrant III, both sine and cosine are negative. We can use the Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\) to find \(\cos\theta\).

Step 4 ：Given that \(\cot \theta = -\frac{3}{4}\) and \(\theta\) is in quadrant IV, we need to find \(\cos \frac{\theta}{2}\).

Step 5 ：We can use the half angle formula for cosine, which is \(\cos\frac{\theta}{2} = \pm\sqrt{\frac{1+\cos\theta}{2}}\). We know the value of \(\cot\theta\), but we need to find the value of \(\cos\theta\).

Step 6 ：Since \(\theta\) is in quadrant IV, cosine is positive and sine is negative. We can use the identity \(\cot^2\theta + 1 = \csc^2\theta\) to find \(\cos\theta\).

Step 7 ：The exact value of \(\sin 2 \theta\) is approximately \(\boxed{0.71}\) and the exact value of \(\cos \frac{\theta}{2}\) is approximately \(\boxed{0.95}\).