Question

Let `theta` be an angle in quadrant II such that `sintheta=4/5`. How do you find the exact values of `sec theta` and `cot theta`?

Answer 1

`sec t = - 5/3`
`cot t = - 3/4`

Explanation:

`sin t = 4/5`. Find cos t
`cos^2 t = 1 - sin^2 t = 1 - 16/25 = 9/25` --> `cos t = +- 3/5`
`cos t = - 3/5` (because t is in Quadrant 2)
`sec t = 1/(cos t) = - 5/3`
`cot t = cos t/(sin t) = (-3/5)(5/4) = - 3/4`

Answer 2

`sectheta=-5/3,cottheta=-3/4`

Explanation:

`"begin by finding " costheta`

`•color(white)(x)costheta=+-sqrt(1-sin^2theta)`

`color(white)(xxxxxx)=+-sqrt(1-16/25)=+-3/5`

`"since " theta " is in second quadrant then"`

`costheta=-3/5`

`•color(white)(x)sectheta=1/costheta=1/(-3/5)=-5/3`

`•color(white)(x)cottheta=costheta/sintheta=(-3/5)/(4/5)=-3/5xx5/4=-3/4`