Question

sin theta = -sqrt(2)/2 So theta = ? or ?

I know the answers are 225 degrees and 315 degrees since the reference angle is 45 degrees and it's negative. But, how do I figure this out without memorizing the values?

Answer 1

look at a cheat sheet

Answer 2

Yes, I can do that :) But, I'd like to know how to derive it.

Answer 3

memorize this triangle

Answer 4

No... How do I get that from \[\sin \theta - \frac{ \sqrt{2} }{ 2 }\]

Answer 5

sine is negative, so the corresponding points on the unit circle are below the x axis (since sine is the second coordinate)
put the triangle above in the unit circle like this

Answer 6

@satellite73 There we go. That's what I was looking for. I knew it was the 45 45 90 triangle, but I just couldn't see it. So how is that leg the square root of 2 DIVIDED by 2? I thought the measurements were 1 for both legs and just the square root of 2 for the hypotenuse.

Answer 7

ratios are all the same no matter how you label the triangle
BUT!!!
you should always have the hypotenuse of 1, so that the triangle gets placed inside the unit circle with the hypotenuse as the radius
and the radius of the unit circle is 1, hence the name

Answer 8

do don't label the triangle as \(1:1:\sqrt2\) but instead use \(\frac{\sqrt2}{2}:\frac{\sqrt2}{2}:1\)

Answer 9

Oh! That makes sense!

Answer 10

we can derive the lengths of the legs for this triangle if you like

Answer 11

Nope. Just did that in my head. Thanks!

Answer 12

or you can just take the triangle you know and divide all sides by \(\sqrt2\) to get that one
ok

Answer 13

I just need to remember the 45 45 90 triangle with the measurements like that from now on.

Answer 14

yeah and same for the 30 - 60 - 90 triangle, make the hypotenuse 1

Answer 15

Oh, wow. Ok. Yeah that would help too. Divide all the measurements by 2 in that case. Neat.