Question

How do you use the quadratic formula to solve `sec^2theta-2sectheta-4=0` for `0<=theta<360`?

Answer 1

`theta=72^@` or `144^@` or `216^@` or `288^@`

Explanation:

The solution of quadratic equation `ax^2+bx+c=0` is `x=(-b+-sqrt(b^2-4ac))/(2a)`

Hence using quadratic formula as `sec^2theta-2sectheta-4=0`

`sectheta=(-(-2)+-sqrt((-2)^2-4xx1xx(-4)))/2`

= `(2+-sqrt(4+16))/2=(2+-sqrt20)/2=(2+-2sqrt5)/2=1+-sqrt5`

As `sqrt5=2.23607`, `sectheta=3.23607` or `-1.23607`

If `sectheta=3.23607`, `theta=72^@` or `360^@-72^@=288^@`

and if `sectheta=-1.23607`, `theta=144^@` or `360^@-144^@=216^@`

Note : `sec72^@=sqrt5+1` and `sec144^@=-sqrt5+1`