How do you find all solutions of the equations $sec^2x-secx=2$ in the interval $[0,2pi)$ ?

Question

13226 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-11-20T13:37:10

There are three solutions to this equation:

$x_1 = \pi/3; \qquad x_2 = (5\pi)/3;$ and $x_3 = \pi;$

Setting $t=\secx$ , we get the following quadratic equation:

$t^2-t-2=0$

which has as solutions, $t_1 = 2; \qquad t_2 = -1;$

So we need to look for solutions to the equations -
[1] $\secx = 2 : \qquad \cosx = 1/2 \rightarrow x_1 = \pi/3; \quad x_2 = (5\pi)/3;$

[2] $\secx = -1: \qquad \cosx=-1 \rightarrow x_3 = \pi;$

How do you find all solutions of the equations sec^2x-secx=2sec^2x-secx=2 in the interval [0,2pi)[0,2pi)?