Question

Find all solutions of the equation in the interval [0,2π) sec(theta) +2=0

Answer 1

I will first reveal the most obvious secret of trig: Most of the problems are 30/60/90 or 45/45/90 triangles of some sort.

$\sec \theta + 2 = 0$

$\dfrac{1 }{ \cos \theta} = -2$

$\cos \theta = - \frac 1 2$

The rule to remember is $\cos x = \cos a$ has solutions

$x = \pm a + 2 \pi k \quad$ integer $k$

Continuing where we left off, a cosine of -1/2 is a 30/60/90 triangle in the second or third quadrant; we pick second, and a little thought tells us the angle is  $120^\circ$.

$\cos \theta = - \frac 1 2$

$\cos \theta = \cos \frac {2\pi} 3$

$\theta = \pm \frac {2\pi} 3 + 2 \pi k \quad$ integer $k$

In the range we want, that's $k=0$ with the plus sign and $k=1$ with the minus, so 

$\theta = \frac {2\pi} 3$ or $-\frac {2\pi} 3 + 2 \pi = \frac{4 \pi} 3$

$\theta = \frac {2\pi} 3$ or $\frac{4 \pi} 3$