Question

How do you solve `sec^2x+0.5tanx=1` and find all solutions in the interval `[0,2pi)`?

Answer 1

`0, 153^@43, 180^@, 333^@43`

Explanation:

Call tan x = t. Apply the trig identity:
`1 + t^2 = sec^2 x`
The equation is transformed to:
`1 + t^2 + t/2 = 1`
t^2 + t/2 = 0
t(t + 1/2) = 0
There are 2 cases:
a. t = tan x = 0 --> arc x = 0 and arc x = pi
Calculator gives:
b. tan x = -1/2 --> arc `x = - 26^@57` and `x = -26.57 + 180 = 153^@43`.
The co-terminal arc of `x = -26^@57` is: arc `x = 333^@43`
Answers for `(0, 360)`:
`0, 153^@43, 180^@, 333^@43`