Question

How do you use `csctheta=5` to find `cottheta`?

Answer 1

`cottheta=+-2sqrt6`

Explanation:

The terminal ray of an angle `theta` in standard position intersects a point `(x,y)` on the unit circle such that `csctheta=1/y`, `cottheta=x/y`, and `x^2+y^2=1`.

Knowing that `csctheta=5` we know that `y=1/5`

Since `x^2+y^2=1`, `x=+-sqrt(1-y^2)`

`cottheta=x/y=(+-sqrt(1-y^2))/y=(+-sqrt(1-(1/5)^2))/(1/5)`
`cottheta=+-5sqrt(1-1/25)=+-5sqrt(24/25)=+-cancel5sqrt24/cancel5=+-2sqrt6`

Answer 2

`+- 2sqrt6`

Explanation:

Use trig identity:
`1 + cot^2 t = csc^2 t`
In this case:
`1 + cot^2 t = 25`
`cot^2 t = 24`
`cot t = +- 2sqrt6`