00:01
All right, so in this problem, we got a sign of a, sign of alpha, is equal to 8 over 17, and the cosine of beta equals 113 over 8 square root of 113.
00:13
And then there's a whole bunch of stuff we have to find.
00:15
But the first thing we need to find is we need to find what alpha is.
00:20
So, alpha, we're told, is between 0 and pi over 2.
00:25
That means it's in quadrant 1.
00:28
So it's going to be here.
00:29
Here's alpha.
00:32
And then when we put in sign is the opposite over the hypotenuse, and when we do the pythagorean theorem, our adjacent side's going to be 15.
00:41
Now, it's important to know it's in quadrant 1 because that 15 is going to be positive.
00:47
All right, when we do the have the cosine of beta, when we use that one, here this is telling us that beta, the angle beta, has to be in quadrant 4 from negative pi over 2 to 0.
01:00
And that's important because if we draw it, it's going to look like this.
01:04
It's going to go down into quadrant 4 like this.
01:08
So beta is right here.
01:11
Cosine is the adjacent.
01:13
So 113.
01:15
It's the adjacent side over 8 square root of 113.
01:19
And when you do the pythagorean theorem and you square, yeah, that's right.
01:28
Did i copy that, right? yeah, when you squared the...
01:35
I'm sorry, i think i copied that one down wrong.
01:39
Give me one second.
01:42
Okay, yes, i did copy that down wrong, so don't mind me.
01:47
This should be eight squared of 113s on top.
01:53
That's smaller.
01:55
Eight squared of 113 is smaller than 113.
01:59
I can't have a bigger leg than a hypotenuse.
02:05
Okay.
02:07
So adjacent is the 8 squared of 113.
02:11
113 is the hypotenuse.
02:14
It's good if you copy down correctly.
02:17
So when we do the pythagorean theorem, 113 squared big number equals 8 squared 64 times 113, so big number.
02:29
And then you subtract.
02:32
That from 113.
02:34
And when you simplify down that square root, our missing side is going to be seven square root of 113.
02:42
But because it's in quadrant 4, this side is negative.
02:45
So this is important because now here we know that the sign of alpha is 8 over 17 and the cosine of alpha is 15 over 17.
03:03
Cosine of alpha is 15 over 17.
03:06
And we know all of our trig ratios.
03:10
It's 8 over 15.
03:12
Even over here, the sign of beta is now negative 7 squared of 113 over 113.
03:22
The cosine of beta is 8 square of 130, over 113 and the tangent of beta is opposite.
03:40
So it'll be negative 7 over 8.
03:43
The square of 113s are going to reduce out.
03:46
All right.
03:46
We need all these because the first thing we want to find is the sign of alpha plus beta.
03:54
Well, there's an identity that says we can, that's just going to equal the sign of alpha.
04:03
Times the cosine of beta plus the sign of beta times the cosine of alpha.
04:20
You switch them.
04:22
Well, we have them.
04:23
They're listed over there.
04:24
The sign of a, so that's 8 over 17 times 8 squared of 113 over 113 plus sine of beta is negative 7, negative 7, square root of 130, negative 7 squared of 113 over 113 times the cosine of alpha, which is 15 over 17.
04:57
Okay, and you'll notice that 17 times 113 is the common denominator right there.
05:05
So we just have to multiply and combine our like terms.
05:10
So on top we have 64 square root of 113 plus a negative 105 squared of 113, right? negative 7 times 15 is 105.
05:24
Combine those 64 and negative 105 comes out to be 41 negative.
05:31
So we get negative 41 squared of 113 over 17 times 113.
05:41
14's prime, 17's prime, 113s prime.
05:44
So if we multiply 17 times 113, we get 1 ,921.
05:54
It's not going to reduce with any of that.
05:56
41's prime.
05:59
So 1 ,921's on the bottom.
06:03
This is our answer, as weird as it is, all simplify down and reduce down for sine of a plus b.
06:12
Okay.
06:13
Now for the sign of a minus b, alpha minus beta, it's the same thing.
06:24
Sign of alpha times the cosine of beta minus the sine of beta minus the sign of beta.
06:33
Let me sure i'm copying it right yet.
06:37
Times the cosine of alpha.
06:40
Well, the numbers are exactly the same except the only thing that changed.
06:44
Was this minus sign.
06:46
So this is still going to come out to be 64 squared of 113.
06:51
But now we're subtracting a negative 105 square root of 113.
06:58
And it's still over our common denominator 17 times squared times 113.
07:04
Now i'm not multiplying it together because i'm hoping something reduces out.
07:09
Those are prime numbers though.
07:11
But now when we add the opposite here, we get one.
07:15
169, which is 13 times 13, but 13 doesn't go into 113.
07:25
I don't believe.
07:30
Yeah, it doesn't.
07:31
So we're going to wind up with 169 because we have to add it times the square root of 113 over our common denominator, 1 ,9...