00:01
For the first one here, we're told that we have sign of x is equal to 7 over 9, and we're in quadrant 2.
00:06
So when we're in quadrant 2, i'm going to go ahead and draw in the reference triangle here.
00:12
That means for this, the opposite is going to be 7, and the hypotenuse is 9.
00:16
So the next steps now is to figure out, well, what is that value of the third piece? so i'm going to do 81 minus 49.
00:21
I get 32, and so the square root of 32 can simplify into 4 radical 2.
00:26
And that's because 32 is 16 times 2.
00:29
So now i can find the sine of 2x, cosine of 2x, and tan of 2x.
00:34
So for sign of 2x, it could becomes 2 and then sign of x, cosine of x.
00:41
So i would do 2 times.
00:43
I already know that the sign of x is 7 over 9, and then i can now do cosine.
00:47
So cosine is going to be 4 times the square to 2 all over 9.
00:51
So in the numerator, we've got 7 times 4, which is 28, and then 28 times 2.
00:56
2 is 56.
00:58
So i have 56 times the square root of 2.
01:01
And in the denominator, 9 times 9 is 81.
01:04
So that would be for sine of 2x.
01:06
For a cosine of 2x, we have a number of options here.
01:09
We can do something like cosine square of x minus sine squared of x, or you can do 2 cosine square of x minus 1 or minus 2 cosine square of x.
01:18
It does not matter.
01:19
So i'm going to do this one here.
01:21
And so from there, cosine squared, as we said before, is 4 square root of 2, all over.
01:26
9.
01:28
I'm going to square that and then i'll do minus sign squared.
01:30
So that would be 7 over 9 squared.
01:32
So the 4 over 2 when i square that out, that's 32 over 81.
01:38
And then this becomes 49 over 81.
01:41
Now it should make sense that it's going to be a negative number because we're dealing with an answer that was originally in quadrant.
01:49
Oh, real quick, real quick.
01:50
Got to back it up a little bit.
01:52
Cosine here is negative because of that value there.
01:54
So this is a negative here.
01:56
Making this in negative 56, radical 2 over 81.
01:59
Okay, so now that we're back on track, the cosine here, that's going to be negative.
02:02
This still squares out to become a positive answer, though.
02:04
So it doesn't really change anything from there.
02:07
But we'll do 32 minus 49, and this is going to get me a negative 17 over 49...