# How do you sketch a right triangle corresponding to costheta=5/7 and find the third side, then find the other five trigonometric functions?

Mar 10, 2017

See below.

#### Explanation:

$\cos \theta$=( adjacent side ) / ( hypotenuse )$= \frac{5}{7}$

Adjacent side$= 5 k$ ; Hypothenuse$= 7 k$

Let opposite side be $x$

By Pythagoras theorem,

$49 {k}^{2} = 25 {k}^{2} + {x}^{2}$

$49 {k}^{2} - 25 {k}^{2} = {x}^{2}$

${x}^{2} = 24 {k}^{2}$

$x = \sqrt{24} k$

$x = 2 \sqrt{6} k$

Third side = $x = 2 \sqrt{6} k$

$\sin \theta = \frac{2 \sqrt{6}}{7}$

$\cos \theta = \frac{5}{7}$

$\tan \theta = \frac{2 \sqrt{6}}{5}$

$\cot \theta = \frac{5}{2 \sqrt{6}}$

$\sec \theta = \frac{7}{5}$

$\cos e c \theta = \frac{7}{2 \sqrt{6}}$