Question

The point `(-5,-2)` is on the terminal side of an angle in standard position, how do you determine the exact values of the six trigonometric functions of the angle?

Answer 1

`sintheta~~-0.37`
`costheta~~-0.93`
`tantheta~~0.40`
`csctheta~~-2.69`
`sectheta~~-1.08`
`cottheta~~2.50`

Explanation:

Here's a diagram I made using Desmos:

The unknown angle `theta` can be calculated by using `tan` of the lengths we already know:

`theta = 180º + tan^-1(2/5)`

`~~180º+21.8º=201.8º`

You can use a calculator or a sine table to calculate the trig functions:

`sintheta~~-0.37`
`costheta~~-0.93`
`tantheta~~0.40`
`csctheta~~-2.69`
`sectheta~~-1.08`
`cottheta~~2.50`

If your calculator doesn't support `tan`, `csc`, `sec`, or `cot`, here are some helpful conversions you can use:

`tantheta=sintheta/costheta`

`csctheta=1/sintheta`

`sectheta=1/costheta`

`cottheta=1/tantheta=1/(sintheta/costheta)=costheta/sintheta`