00:01
Okay, this is a fun little problem.
00:02
We have this triangle.
00:04
We have a, b, and c, and it gives us some measurements and some angles and that type of thing, and we're asked to find four things.
00:13
So, first thing, we're going to find the distance between a and c, okay? so that is number i, and so we're looking for this distance right here, okay? and so what do we know? well, we know in order to find that third distance, we've got two sides.
00:31
So we're going to need to know at least one angle.
00:36
And so when i'm looking at this, the first angle that i see that i can figure out is the angle from right here to that horizontal, i'm sorry, that vertical line pointing north.
00:50
And i know that line is 180.
00:54
180, take away 125, is going to give me an angle of 55 degrees right here.
01:02
And so if i can imagine extending that line down and imagine this right triangle, this is 90 degrees right here, that means 90 degrees plus 55 is 145 and i know that that the whole triangle is 180 so this is by default 35 degrees okay that angle b is 35 degrees and how does that help us well because now we've got two sides and an enclosed angle so in order to find the length of the other side we can use the law of cosines and and the law of cosines, first of all, let me do a little bit of labeling.
01:45
So this is called side b because it's across from angle b.
01:49
This is side a because it's across from angle a.
01:52
And this is side c, okay? so law of cosines says that b squared equals a squared plus c squared minus 2ac times the cosine of angle b, which is 35.
02:07
Let me make sure my calculator is in degree mode.
02:13
So to find the length of side b, i'm just going to take the square root of a squared plus c squared minus 2ac cosine 35.
02:21
And you can just plug that whole thing into your calculator.
02:25
In fact, you know what i should have done? let's put the actual numbers in here.
02:30
Let's just erase that and start over.
02:33
Okay, so that's going to be the square root of 460 squared plus 430 squared minus 2 times 460 times 430 times the cosine of 35.
02:49
Okay, and i'm going to put that in my calculator just like that.
02:52
460 squared plus 430 squared minus 2 times 460 times 430 times the cosine of 35.
03:04
And i get that the side length b is 269 .15.
03:10
We'll just do it to the hundredths place.
03:11
So this is 269 .15.
03:14
Okay, so we've answered the first question, 269 .15.
03:21
All right, now we're asked to find the measure of angle acb, which is the top angle in that triangle, okay? and so now i'm going to switch over to the law of sines.
03:35
And law of sines says that the sine of an angle over its opposite side length, which would be side length, whoops, b, which is 269 .15, is equal to the sine of, we're just going to say angle c over the side opposite that, which is 430.
04:02
Okay, i want the sine of angle c by itself, and so i'm going to say, i'm going to multiply both sides by 430...