Question

How are the 6 basic trigonometric functions related to right triangles?

Answer 1

The six basic trigonometric functions are:
1. Sine, `sintheta`
2. Cosine, `costheta`
3. Tangent, `tantheta`
4. Cotangent, `cottheta`
5. Secant, `sectheta`
6. Cosecant, `csctheta`

Take the following triangle for example:

Let the angle marked at A be `theta`.
The longest side of the triangle is the hypotenuse, the side next to the angle is the adjacent and the side opposite to it is the opposite.

Then,

`sintheta =(opp)/( hyp)=a/h`

`costheta =(adj)/( hyp)=b/h`

`tantheta =(opp)/(adj)=a/b=(sintheta)/(costheta)`

`cottheta =(adj)/(opp)=b/a=(costheta)/(sintheta)=1/tantheta`

`sectheta =(hyp)/(adj)=h/b=1/costheta`

`csctheta =(hyp)/(opp)=h/a=1/sintheta`

Now if we the triangle were to have sides 3, 4 and 5 units long such that
`a=4, b=3, h=5`
The six basic trigonometric functions would be:

`sintheta =a/h=4/5`

`costheta=b/h=3/5`

`tantheta =a/b=4/3`

`cottheta =b/a=3/4`

`sectheta =h/b=5/3`

`csctheta =h/a=5/4`