How do you find the exact value of the six trigonometric functions of the angle whose terminal side passes through (3,-4)?

Apr 27, 2017

See explanation.

Explanation:

To find the values of all the trigonometric functions first you have to find the distance between the point on the angle's side and the origin.

Here the distance is:

$r = \sqrt{{\left(3 - 0\right)}^{2} + {\left(- 4 - 0\right)}^{2}} = \sqrt{{3}^{3} + {4}^{2}} = \sqrt{25} = 5$

Now we can calculate the functions:

$\sin \alpha = \frac{y}{r} = \frac{- 4}{5} = - 0.8$

$\cos \alpha = \frac{x}{r} = \frac{3}{5} = 0.6$

$\tan \alpha = \frac{y}{x} = \frac{- 4}{3} = - 1 \frac{1}{3}$

$\cot \alpha = \frac{x}{y} = \frac{3}{-} 4 = - \frac{3}{4}$

$\sec \alpha = \frac{r}{x} = \frac{5}{3} = 1 \frac{2}{3}$

$\csc \alpha = \frac{r}{y} = \frac{5}{-} 4 = - \frac{5}{4} = - 1 \frac{1}{4}$