# Question #e4d1e

Mar 14, 2017

You just need to simplify the equation.
Here are the steps, from the beginning to the second line of your problem.

#### Explanation:

You have the following equation:
${\sin}^{2} \theta - 2 \sin \theta - 1 = 0$

Note that ${\sin}^{2} \theta = {\left(\sin \theta\right)}^{2}$,
so let's say $\sin \theta = x$ and rewrite the equation.

We get:
${x}^{2} - 2 \cdot x - 1 = 0$

This is a quadratic equation that you should know how to solve, using the (in)famous quadratic formula:
$x = \frac{- b \setminus \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$
where $a$, $b$, and $c$ are the coefficients of the second degree polynomial:
$a {x}^{2} + b x + c = 0$

In our case,
we readily see that we have
$a = 1$,
$b = - 2$, and
$c = - 1$

So plugging-in these numbers in the quadratic formula gives the following:
$x = \frac{+ 2 \setminus \pm \sqrt{{\left(- 2\right)}^{2} - 4 \cdot 1 \cdot \left(- 1\right)}}{2 \cdot 1} = \frac{2 \setminus \pm \sqrt{4 + 4}}{2}$
so
$x = \frac{2 \setminus \pm \sqrt{8}}{2}$
(Note that this is representing the two solutions of the equation with ${x}_{1} = \frac{2 + \sqrt{8}}{2}$ and ${x}_{2} = \frac{2 - \sqrt{8}}{2}$ written together with the $\setminus \pm$ ("plus or minus" sign) ).

Replacing the x back to its original form:
$\sin \theta = \frac{2 \setminus \pm \sqrt{8}}{2}$

So here we are at the second line. I hope this helped.

Mar 14, 2017

${\sin}^{2} \theta - 2 \sin \theta - 1 = 0$

We know that the roots of general quadratic equation of the form $a {x}^{2} + b x + c = 0$ are

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

Applying this to our quadratic equation of $\sin x$ we get

$\sin x = \frac{2 \pm \sqrt{{\left(- 2\right)}^{2} - 4 \cdot 1 \cdot \left(- 1\right)}}{2 \cdot 1}$

$\implies \sin x = \frac{2 \pm \sqrt{8}}{2}$

$\implies \sin x = \frac{2 \pm \sqrt{2 \cdot {2}^{2}}}{2}$

$\implies \sin x = \frac{2 \pm 2 \sqrt{2}}{2}$

$\implies \sin x = 1 \pm \sqrt{2}$

As $\sin x = 1 + \sqrt{2} > 0 \text{ not possible}$

we have

$\implies \sin x = 1 - \sqrt{2} = - 0.414 \approx - \sin {24}^{\circ}$

So either $\sin x = \sin \left(180 + 24\right) = \sin {204}^{\circ}$

$\implies \sin x = {204}^{\circ}$

Or

$\sin x = \sin \left(360 - 24\right) = \sin {336}^{\circ}$

$\implies x = {336}^{\circ}$