# If sin=3/5 what is cos,tan,csc, sec,cot?

May 6, 2018

color(brown)(csc theta = 5/3

color(blue)(cos theta = 4/5, sec theta = 5/4

color(green)(tan theta = 3 / 4, cot theta = 4/3

#### Explanation:

$\sin \theta = \text{(Oppo. side) /( Hypotenuse)} = \frac{3}{5}$

According to Pythagorus Theorem,

$\text{Adj. side } A B = \sqrt{{\left(A C\right)}^{2} - {\left(B C\right)}^{2}} = \sqrt{{5}^{2} - {3}^{2}} = 4$

Any triangle whose sides are in the ratio 3:4:5 is a right triangle.

Such triangles that have their sides in the ratio of whole numbers are called color(red)("Pythagorean Triples."

There are an infinite number of them, and this is just the smallest.

$\csc \theta = \frac{1}{\sin} \theta = \frac{5}{3}$

$\cos \theta = \frac{A B}{A C} = \frac{4}{5}$

$\sec \theta = \frac{1}{\cos} \theta = \frac{5}{4}$

$\tan \theta = \frac{B C}{A B} = \frac{3}{4}$

$\cot \theta = \frac{1}{\tan} \theta = \frac{4}{3}$