Question

What are the solutions to 10(sec^2)x+5(tan^2)x-15=0 over the interval (0,2pi)?

Answer 1

`pi/6; (2pi)/3; (7pi)/6; (5pi)/3`

Explanation:

`10sec^2 x + 5tan^2 x - 15 = 0`
Replace `sec^2 x` by `(1 + tan^2 x)` from trig identity.
`10(1 + tan^2 x) + 5tan^2 x - 15 = 0`
`10tan^2 x + 5tan^2 x - 5 = 0`
`15tan^2 x = 5` --> `tan^2 x = 1/3`
`tan x = +- sqrt3/3`
Trig table and unit circle give 4 solutions for tan x -->
a. `tan x = sqrt3/3` -->
`x = pi/6` and `x = pi/6 + pi = (7pi)/6`
b. `tan x = - sqrt3/3` -->
`x = (2pi)/3` and `x = pi + (2pi)/3 = (5pi)/3`