# Without solving, how do you determine the number of solutions for the equation sectheta=(2sqrt3)/3 in -180^circ<=theta<=180^circ?

Feb 18, 2018

Two solutions.
See explanation.

#### Explanation:

$\sec \theta = \frac{1}{\cos \theta} = \left(\frac{2 \sqrt{3}}{3}\right)$

So you're really looking at where
$\cos \left(\theta\right) = \frac{3}{2 \sqrt{3}} = \frac{\sqrt{3}}{2}$ (looks more familiar?)

$- {180}^{\circ} = - \pi$ radian
${180}^{\circ} = \pi$ radian

Since the cosine function is "even", $\cos \left(- x\right) = \cos \left(x\right)$ for all $x$.
So if there is a solution from 0 to $\pi$, there is also a solution from 0 to $- \pi$.

We can readily see that $0 < \frac{\sqrt{3}}{2} < 1$,
because $\sqrt{3} < \sqrt{4}$.
We know that the range of the cosine function is between -1 and 1, so this is good news, this means that there is a solution (if the value was larger than 1, we would have had zero solution).

Therefore,
$\cos \left(\theta\right) = \frac{\sqrt{3}}{2}$ has a solution that is between (0 and ${180}^{\circ}$), and another solution that is between (0 and $- {180}^{\circ}$).
Thus,
$\sec \theta = \left(\frac{2 \sqrt{3}}{3}\right)$ has two solutions in the range $- {180}^{\circ} < \theta < {180}^{\circ}$.
Q.E.D.