Chapter 5 - Review Exercises - Page 248: 15

$sin~165^{\circ} = \frac{\sqrt{6}-\sqrt{2}}{4}$ $cos~165^{\circ} = -(\frac{\sqrt{2}+\sqrt{6}}{4})$ $tan~165^{\circ} = \sqrt{3}-2$ $csc~165^{\circ} = \sqrt{6}+\sqrt{2}$ $sec~165^{\circ} = \sqrt{2}-\sqrt{6}$ $cot~165^{\circ} = -(\sqrt{3}+2)$

$sin~165^{\circ} = sin(120^{\circ}+45^{\circ})$ $sin~165^{\circ} = sin~120^{\circ}~cos~45^{\circ}+cos~120^{\circ}~sin~45^{\circ}$ $sin~165^{\circ} = (\frac{\sqrt{3}}{2})(\frac{\sqrt{2}}{2})+(-\frac{1}{2})(\frac{\sqrt{2}}{2})$ $sin~165^{\circ} = \frac{\sqrt{6}}{4}-\frac{\sqrt{2}}{4}$ $sin~165^{\circ} = \frac{\sqrt{6}-\sqrt{2}}{4}$ $cos~165^{\circ} = cos(120^{\circ}+45^{\circ})$ $cos~165^{\circ} = cos~120^{\circ}~cos~45^{\circ}-sin~120^{\circ}~sin~45^{\circ}$ $cos~165^{\circ} = (-\frac{1}{2})(\frac{\sqrt{2}}{2})-(\frac{\sqrt{3}}{2})(\frac{\sqrt{2}}{2})$ $cos~165^{\circ} = -\frac{\sqrt{2}}{4}-\frac{\sqrt{6}}{4}$ $cos~165^{\circ} = -(\frac{\sqrt{2}+\sqrt{6}}{4})$ $tan~165^{\circ} = tan(120^{\circ}+45^{\circ})$ $tan~165^{\circ} = \frac{tan~120^{\circ}+tan~45^{\circ}}{1-tan~120^{\circ}~tan~45^{\circ}}$ $tan~165^{\circ} = \frac{(-\sqrt{3})+(1)}{1-(-\sqrt{3})(1)}$ $tan~165^{\circ} = \frac{1-\sqrt{3}}{1+\sqrt{3}}\cdot \frac{1-\sqrt{3}}{1-\sqrt{3}}$ $tan~165^{\circ} = \frac{4-2\sqrt{3}}{-2}$ $tan~165^{\circ} = \sqrt{3}-2$ $csc~165^{\circ} = \frac{1}{sin~165^{\circ}}$ $csc~165^{\circ} = \frac{1}{\frac{\sqrt{6}-\sqrt{2}}{4}}$ $csc~165^{\circ} = \frac{4}{\sqrt{6}-\sqrt{2}}\cdot \frac{\sqrt{6}+\sqrt{2}}{\sqrt{6}+\sqrt{2}}$ $csc~165^{\circ} = \frac{4(\sqrt{6}+\sqrt{2})}{4}$ $csc~165^{\circ} = \sqrt{6}+\sqrt{2}$ $sec~165^{\circ} = \frac{1}{cos~165^{\circ}}$ $sec~165^{\circ} = -\frac{1}{(\frac{\sqrt{2}+\sqrt{6}}{4})}$ $sec~165^{\circ} = -\frac{4}{\sqrt{2}+\sqrt{6}}\cdot \frac{\sqrt{6}-\sqrt{2}}{\sqrt{6}-\sqrt{2}}$ $sec~165^{\circ} = -\frac{4(\sqrt{6}-\sqrt{2})}{4}$ $sec~165^{\circ} = \sqrt{2}-\sqrt{6}$ $cot~165^{\circ} = \frac{1}{tan~165^{\circ}}$ $cot~165^{\circ} = \frac{1}{\sqrt{3}-2}\cdot \frac{\sqrt{3}+2}{\sqrt{3}+2}$ $cot~165^{\circ} = \frac{\sqrt{3}+2}{-1}$ $cot~165^{\circ} = -(\sqrt{3}+2)$