# How do you find the reference angle for -515 degrees?

Dec 13, 2017

$205$ degrees

#### Explanation:

$- 515$ is no the simplest form for this angle.

See, $900$ degrees is the same angle as $180$ degrees, and one of the ways I like to check that that's true -- if I have a calculator handy --is to plug in $\sin \left(180\right)$ and $\sin \left(900\right)$, and if they are the same angle, they'll give me the same answer. And they do, $0$ for both of them.

Let's try and find the simplified angle for $- 515$. Now, that negative might look scary, but it just means that the angle was found by going clockwise, unlike the normal way where we move counter-clockwise.

So we'll ignore the negative sign for now. The first thing I do is imagine drawing a line all the way around the circle

That used up $360$ degrees. Now we have $115$ degrees left

Now I move another $90$ degrees

Now I have $65$ degrees left

Now we can look at the final picture and see how many degrees it would take to reach this angle, but going in a clockwise direction . That'll give us our reference angle

So, we'll need to go $180$ degrees. And we know that we used up $65$ degrees, so there are $25$ degrees left in that qudarant. $180 + 25$ gives us $205$

So, we are saying that $205$ and $- 515$ are two different ways to reference the same angle. Let;s plug them into $\sin \left(\theta\right)$ and find out!

$\sin \left(205\right) = - 0.4226$
$\sin \left(- 515\right) = - 0.4226$

Yep! The reference angle for $- 515$ is $205$