1. **Problem 5:** Given $\cos \theta = -\frac{3}{4}$ and $\theta$ is in quadrant II, find the other five trigonometric ratios. 2. **Step 1: Understand the quadrant and sign rules.** - In quadrant II, $\sin \theta > 0$, $\cos \theta < 0$, $\tan \theta < 0$. 3. **Step 2: Use the Pythagorean identity to find $\sin \theta$.** $$\sin^2 \theta + \cos^2 \theta = 1$$ $$\sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(-\frac{3}{4}\right)^2 = 1 - \frac{9}{16} = \frac{7}{16}$$ $$\sin \theta = +\sqrt{\frac{7}{16}} = \frac{\sqrt{7}}{4}$$ (positive in quadrant II) 4. **Step 3: Find $\tan \theta$.** $$\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{\sqrt{7}}{4}}{-\frac{3}{4}} = -\frac{\sqrt{7}}{3}$$ 5. **Step 4: Find the reciprocal functions:** - $\csc \theta = \frac{1}{\sin \theta} = \frac{4}{\sqrt{7}} = \frac{4\sqrt{7}}{7}$ after rationalizing. - $\sec \theta = \frac{1}{\cos \theta} = \frac{1}{-\frac{3}{4}} = -\frac{4}{3}$. - $\cot \theta = \frac{1}{\tan \theta} = -\frac{3}{\sqrt{7}} = -\frac{3\sqrt{7}}{7}$ after rationalizing. --- 6. **Problem 6:** Determine the exact value of each expression. **a)** $\sin 45^\circ - \cos 45^\circ$ - $\sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2}$ - So, $\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} = 0$ **b)** $(\csc 30^\circ)^2$ - $\sin 30^\circ = \frac{1}{2}$, so $\csc 30^\circ = 2$ - Square: $2^2 = 4$ **c)** $(\sin (\frac{7\pi}{6}))(\csc (\frac{7\pi}{6}))$ - $\sin \frac{7\pi}{6} = -\frac{1}{2}$ - $\csc \frac{7\pi}{6} = \frac{1}{\sin \frac{7\pi}{6}} = -2$ - Product: $-\frac{1}{2} \times -2 = 1$ **d)** $\csc^2(135^\circ) - \cot^2(135^\circ)$ - Use identity: $\csc^2 \theta - \cot^2 \theta = 1$ - So, answer is $1$ **e)** $\sin^2 (\frac{2\pi}{3}) + \cos^2 (\frac{2\pi}{3})$ - By Pythagorean identity, this equals $1$ **f)** $(\tan 60^\circ)(\cos 30^\circ) - \sin 60^\circ$ - $\tan 60^\circ = \sqrt{3}$ - $\cos 30^\circ = \frac{\sqrt{3}}{2}$ - $\sin 60^\circ = \frac{\sqrt{3}}{2}$ - Compute: $\sqrt{3} \times \frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{2} = \frac{3}{2} - \frac{\sqrt{3}}{2} = \frac{3 - \sqrt{3}}{2}$ **Final answers:** - 5: $\sin \theta = \frac{\sqrt{7}}{4}$, $\tan \theta = -\frac{\sqrt{7}}{3}$, $\csc \theta = \frac{4\sqrt{7}}{7}$, $\sec \theta = -\frac{4}{3}$, $\cot \theta = -\frac{3\sqrt{7}}{7}$ - 6a: $0$ - 6b: $4$ - 6c: $1$ - 6d: $1$ - 6e: $1$ - 6f: $\frac{3 - \sqrt{3}}{2}$