Question

Question #ec1c9

Answer 1

`2sin(5x)=-sqrt2`

Lets isolate our `sine`:

`2sin(5x)=-sqrt2`

divide by `2` on both sides
`sin(5x)=-sqrt(2)/2`

So, we are looking for a number that, when multiplied by `5`, will give us the ratio of `-sqrt(2)/2`. That sounds like a lot, but let's break it down.

First, the value `-sqrt(2)/2` corresponds to `45^0` or `pi/4`.

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

`5*9=45`

`x=9`

`sin(5*9)=-sqrt(2)/2`
`sin(45)=-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 `-sqrt(2)/2` when plugged into sin(x).

So, when does `sin(x)=-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, `sqrt(2)/2`, from any `Sin(x)` or `Cos(x)` is `pi/4`

Finally, `pi/4` in the 3rd and 4rth quadrants are:
3rd: `pi+pi/4= 5pi/4`
4rth: `2pi-pi/4= 7pi/4`

so `x=5pi/4+-2pi`
and `x=7pi/4+-2pi`

we have "`+-2pi`" at the end because every `2pi` is the cycle around the unit circle, resulting in the same answer.