00:01
This problem, we're told that the sign of alpha is 5 over 13 and that the tangent of beta is negative root 3.
00:05
Now, we're told that alpha is in between negative 3 pi over 2 and negative pi.
00:09
Well, if you think about your quadrants here, remember negative angles go in the reverse direction.
00:15
So negative pi would be back 180, and negative 3 pi over 2 would be back 270 degrees.
00:20
So all that's telling us is that alpha is a second quadrant angle.
00:24
So now we can go ahead and draw the triangle so we can have alpha here.
00:28
So now we know that sign is 5 over 3 .3.
00:30
13.
00:30
So that's the opposite side, which is 5, and the hypotenuse would be 13.
00:34
So we just need to find the adjacent side.
00:37
So to do this, we're going to use the pythagorean theorem.
00:39
So if x squared plus 5 squared equals 13 squared, well, 5 squared is equal to 25, so x squared plus 25 equals 13 squared, which is 169.
00:49
So then we'll subtract 25 from both sides.
00:52
So if x squared equal to 144, and then we'll take the square root of both sides and the square with the 144 is 12.
00:58
Now, keep in mind, because we're in the second quadrant that this x value will be negative.
01:03
So be negative 12.
01:04
So now we can go ahead and draw our angle for beta.
01:07
Notice that beta is in between pi over two and pi, which again, also has a second quadrant angle.
01:13
So in this case, we're told the tangent is negative root 3.
01:16
So that would be the adjacent side, which is going to be root 3 over the adjacent side, which is going to be negative 1.
01:22
So now, again, we can find our hypotenuse.
01:25
So i'm going to call this y.
01:26
So we're going to have the square of the 3 being squared plus negative negative 1 squared equals to y squared.
01:33
We'll root 3 squared is equal to 3.
01:35
Negative 1 squared is 1.
01:36
So we'll have 3 plus 1, which is 4.
01:38
And then to solve for y, we'll take the square root of both sides, and we'll find that y is equal to 2.
01:43
So we'll go ahead and add that to our picture.
01:48
So perfect.
01:48
Now that we have all of our sides for each of these triangles, we can go ahead and find the values for each of the problems that we have.
01:55
So the first one, we have the sign of alpha plus beta.
02:00
Well, we have a formula to do this.
02:02
It says it's the sign of alpha plus the, or sorry, times the cosine of beta.
02:09
Get it ahead of myself.
02:11
And then we're going to add this to the cosine of alpha times the sign of beta.
02:15
All right.
02:15
Well, the sign of alpha, that's opposite over hypotenuse.
02:18
So that'd be 5 over 13.
02:21
Let's put the 13 there.
02:23
And then we're going to multiply it by the cosine of beta.
02:25
So that's the adjacent over hypotenuse, so negative 1 over 2.
02:29
Well, then we're going to add this to the cosine of alpha.
02:31
So that's adjacent over hypotenuse, so negative 12 over 13 times the sine of beta.
02:36
So that would be root 3 over 2.
02:39
So now we just want to simplify.
02:41
Well, 5 over 13 times negative 1⁄2 is negative 5 over 26.
02:45
Then we have negative 12 over 13 times 3 over 2, which is negative, or so we're going to add negative 12 root 3 all over 26.
02:56
And now, because they're like denominators, we can combine these two fractions.
02:59
So our final answer will be negative 5 minus 12 root 3, all over 26.
03:05
So that's the first answer.
03:07
Now, for the second one, we're being asked to find the cosine of alpha plus beta.
03:12
Well, we have a formula for this...