Question #ec1c9

Apr 18, 2017

$2 \sin \left(5 x\right) = - \sqrt{2}$

Lets isolate our $\sin e$:

$2 \sin \left(5 x\right) = - \sqrt{2}$

divide by $2$ on both sides
$\sin \left(5 x\right) = - \frac{\sqrt{2}}{2}$

So, we are looking for a number that, when multiplied by $5$, will give us the ratio of $- \frac{\sqrt{2}}{2}$. That sounds like a lot, but let's break it down.

First, the value $- \frac{\sqrt{2}}{2}$ corresponds to ${45}^{0}$ or $\frac{\pi}{4}$.

I like to use degrees :)
So, we need to find a number that, multiplied by $5$, equals $45$.

$5 \cdot 9 = 45$

$x = 9$

$\sin \left(5 \cdot 9\right) = - \frac{\sqrt{2}}{2}$
$\sin \left(45\right) = - \frac{\sqrt{2}}{2}$

EDIT: I do believe that it is a mistake to use degrees (you can but you need to inlcude more work from the pi/180 conversion factor). Basically the answer should be in radians.

So, we're looking for a number that, when multiplied by 5, equals $- \frac{\sqrt{2}}{2}$ when plugged into sin(x).

So, when does $\sin \left(x\right) = - \frac{\sqrt{2}}{2}$?
Well, sin(x) is negative in the 3rd and 4rth Quadrants. (This is something you kinda have to memorize, though there are patterns that will help you.)

Next, $\frac{\sqrt{2}}{2}$, from any $S \in \left(x\right)$ or $C o s \left(x\right)$ is $\frac{\pi}{4}$

Finally, $\frac{\pi}{4}$ in the 3rd and 4rth quadrants are:
3rd: $\pi + \frac{\pi}{4} = 5 \frac{\pi}{4}$
4rth: $2 \pi - \frac{\pi}{4} = 7 \frac{\pi}{4}$

so $x = 5 \frac{\pi}{4} \pm 2 \pi$
and $x = 7 \frac{\pi}{4} \pm 2 \pi$

we have "$\pm 2 \pi$" at the end because every $2 \pi$ is the cycle around the unit circle, resulting in the same answer.