00:01
So in this problem, we're told that the sign of x is equal to negative four -fifths, and it's in the fourth quadrant.
00:05
So what we want to do is we want to first find the sign of 2x.
00:09
Well, our double angle formula for sign is that it's 2 times the sign of x times the cosine of x.
00:16
Okay.
00:17
Well, in order to use this, we're going to need the cosine.
00:20
Well, knowing that our angle is in the fourth quadrant, we can go ahead and draw this.
00:25
So the sign, that's the opposite side over the hypotenuse, this would be our angle x.
00:32
So now we can use the pythagorean theorem to solve for our adjacent side.
00:36
So we'll have, i'll call it y, i guess, y squared plus negative 4 squared equal to 5 squared.
00:42
Well, negative 4 squared is 16, 5 squared is 25.
00:46
So we'll subtract 16 from both sides, so y square will equal to 9.
00:50
And then we'll take the square root of both sides, so y will equal the 3.
00:54
So now that we have all the size of our triangle, we can go ahead and find the double angle now.
00:59
So we have 2, the sign of x, that would be negative 4 fifths, and a cosine that would be the adjacent over the hypotenuse, that would be 3 5ths.
01:07
And now we can go ahead and multiply.
01:09
So again, you can put 2 over 1.
01:10
2 times 4 is negative 8.
01:13
Negative 8 times 3 is equal to negative 24.
01:16
And for the denominator, we have 1 times 5 times 5, which is 25.
01:20
So negative 24 over 25 is the sign of 2x.
01:24
All right.
01:26
Next, we're going to find the cosine of 2x.
01:29
Well, there's multiple formulas we can use, but i'm going to use 1 minus 2 times the sine squared of x.
01:36
And the reason why i do this is because remember, we were given sine squared.
01:39
It's negative 4 5ths...