# How do you find the remaining trigonometric functions of theta given tantheta=a/b where a and b are both positive?

May 24, 2017

#### Explanation:

As $\tan \theta = \frac{a}{b}$, $\cot \theta = \frac{1}{\frac{a}{b}} = 1 \times \frac{b}{a} = \frac{b}{a}$

$\sec \theta = \pm \sqrt{1 + {\tan}^{2} \theta} = \pm \sqrt{1 + {a}^{2} / {b}^{2}} = \pm \frac{\sqrt{{a}^{2} + {b}^{2}}}{b}$

$\cos \theta = \pm \frac{b}{\sqrt{{a}^{2} + {b}^{2}}}$

$\sin \theta = \tan \theta \times \cos \theta = \pm \frac{a}{b} \times \frac{b}{\sqrt{{a}^{2} + {b}^{2}}} = \pm \frac{a}{\sqrt{{a}^{2} + {b}^{2}}}$

and $\csc \theta = \pm \frac{\sqrt{{a}^{2} + {b}^{2}}}{a}$