# How do you find the exact value of the six trigonometric functions of the angle whose terminal side passes through (-0.5, 4.5)?

Feb 27, 2018

As below.

#### Explanation:

$x = - 0.5 , y = 4.5$

$r = \sqrt{{x}^{2} + {y}^{2}} = \sqrt{{\left(- \frac{1}{2}\right)}^{2} + {\left(\frac{9}{2}\right)}^{2}} = \sqrt{\frac{82}{4}} = \sqrt{\frac{41}{2}}$

$\sin \alpha = \frac{y}{r} = \frac{\frac{9}{2}}{\sqrt{\frac{41}{2}}} = \frac{9}{\sqrt{82}}$

$\cos \alpha = \frac{x}{r} = - \frac{\frac{1}{2}}{\sqrt{\frac{41}{2}}} = - \left(\frac{1}{\sqrt{82}}\right)$

$\tan \alpha = \frac{y}{x} = - \frac{\frac{9}{2}}{\frac{1}{2}} = - 9$

$\csc \alpha = \frac{1}{\sin} \alpha = \frac{\sqrt{82}}{9}$

$\sec \alpha = \frac{1}{\cos} \alpha = - \sqrt{82}$

$\cot \alpha = - \left(\frac{1}{9}\right)$