00:01
So in this problem, we're told that the cosine of angle x is equal to negative 1 over 7, and it's in quadrant 3.
00:06
Our ultimate goal is to find the sign of 2x, the cosine of 2x, and the tangent of 2x.
00:11
Well, in order to find these, we're going to have to solve this triangle.
00:14
So let's go ahead and draw it.
00:16
Well, we know it's in quadrant 3, so we can draw that.
00:18
So this would be our right angle.
00:19
So this would be our angle for x.
00:21
So because we're called the cosine is negative 1 -7, that means the adjacent side would be negative 1, and the hypotenuse would be 7.
00:29
So what we need to do now is find the opposite side.
00:32
To do this, we'll use the quadratic formula.
00:34
So i'm going to call it y.
00:35
So we'll have y squared plus negative 1 squared equal to 7 squared.
00:39
Well, negative 1 squared is positive 1.
00:42
7 squared is 49.
00:43
So i'll subtract 1 from both sides of our equation.
00:46
So have y squared equal to 48.
00:48
So then we'll take the square of the both sides.
00:51
But i know we can simplify the square of the 48 because the perfect square of 16 goes into it three times.
00:56
Well, the square of the 16 is 4, and we'll bring down the root 3.
01:01
So perfect.
01:02
Now we can add this to our picture, but because we're in the third quadrant, this value will have to be negative because it's going down.
01:08
All right.
01:08
So now that we've solved our triangle, we can go ahead and find each of our functions.
01:12
So first, we're going to find the sign of 2x.
01:15
Well, the formula for it is 2 times the sine of x times the cosine of x.
01:20
So in this particular case, if we scroll down, we'll have 2 times the sign of x.
01:25
Well, that's the opposite over the hypotenuse.
01:27
So we'll have negative 4 root 3 over 7.
01:30
And we're going to multiply by the cosine of x, which we already knew was negative 1...