 Trigonometry is useful because it allows us to compute distances from angles, at least if we have a right triangle. But what if we don't? We can still use trigonometry by using what's known as the law of sines or the law of cosines. The problem is this. The basic trigonometric ratios are based around right triangles. But what if we don't have a right triangle? No problem, we can make one. And that's because we can always drop a perpendicular line. You'll go through the details in an assignment. If we do that, we obtain two important relationships between the sides and angles of any triangle. So let's take triangle A, B, C. A useful convention is to label the side lengths with the lowercase version of the opposite vertex. So across from vertex A, I have side length lowercase A. Across from vertex B, I have side length lowercase B. And across from vertex C, I have side length lowercase M, I mean C. And we will refer to the measure of the angle by the capital letters and the length of the side by the lowercase letters. The law of sines uses the sines of the measures of the angles. The sine of A over A is equal to the sine of B over B as well. The sine of B over B is equal to the sine of C over C. And the sine of A over A is equal to the sine of C over C. This is sometimes stated as sine of A over A equals sine of B over B equals sine of C over C. However, it's very bad style to use more than one equals on a line. Don't do it. This isn't one of those rules like remembering to never split an infinitive. It's more of a rule like drive only on the left side of the road, right side of the road. It keeps you from having accidents. The other important relationship is known as the law of cosines. In any triangle A, B, C, C squared equals A squared plus B squared minus 2ab times the cosine of the measure of the angle C. This is like the Pythagorean theorem with the correction factor. So let's see how we can use this. Suppose I have two surveyors standing at points A and B, 200 meters apart. They sign on some object at C and measure the two angles to be 27 degrees and 38 degrees. How far is the object from each of our surveyors? To begin with, we should assume that this is a right triangle because I never gave you something that wasn't. Oh wait, you're confusing me with somebody who's nice. We should check to see if we have a right triangle by finding that third angle. So here the thing to remember is that the sum of the three angles in a triangle is 180 degrees. And so you know two angles, 27 and 38, whatever that third angle is, the three will add to 180 degrees. And so that gives us an equation. And we can solve that equation. And we notice that C is not a right triangle, and so we've wasted time. Or have we? Remember, knowledge is power. We went through all of this trouble to find the measure of angle C. Let's record it because if it's not written down, it didn't happen. Now, since we want to know A and B, we can use the law of signs. So if we look at the law of signs and compare it to our triangle and what we know, we see that if we want to find lower case A, this equation requires us to also know lower case B, which we don't already know, so we can't use it. This equation doesn't even involve lower case A, so it's not relevant. So that means if we have any hope of finding lower case A, we have to use this third equation, sine of A over A equals sine of C over C. So let's fill in our numbers. A has a measure of 27 degrees, C has a measure of 115 degrees, lower case C has a length of 200, and A is the only thing we don't know. Let's rearrange our equation a little bit and solve for A, and we'll use a calculator to find the values of sine of 27 degrees and the sine of 115 degrees, and that gives us our value of A. How about our value of B? Well, this time we have a choice. In the first equation, now we know the value of lower case A and the angles A and B, so we could use that first equation. But we also could use the second equation because we know the measure of the angle B and the measure of the angle C as well as the length C. So which equation do we want to use? And here's a good rule in life. Use the values to your most confident of. What does that mean? Well, the angles 27 and 38 were given, and so we have to assume that these values are correct. If they're not correct, there's nothing we can do about it. Likewise, this length 200 meters, we also have to assume is correct because if it isn't correct, we can't solve the problem. And so that means these angles A and B and this length C are all values we should be pretty confident of. Now, we just calculated two other values, the measure of the angle C, which we found by solving this equation and the measure of length A, which we found by solving this equation. And so the question is, which solution are we more confident in? Now, this is a more complicated equation and it involves a couple of new things, this sine of 27 degrees and so on. And so while we should have some confidence that this is the correct value, we might contrast it with this equation, which is a much simpler equation using things that are very familiar to us. And so we should have more confidence that this value is correct than this value. So what that means is we'll want to use this equation to find the value of B. So substituting in our values, then solving for B, then using a calculator to find the sine of 38 degrees and the sine of 115 degrees gives us our value for B. For example, suppose we have a triangle like the one shown, we can label the unknown side little c, so the opposite angle is going to be big C. We can call the other two sides little a and little b, which means the opposite angles are big a, big b. And we can try to use the law of sines. Let's see, this first equation requires us to have angles a and b and side lengths a and b. And while we have the side lengths, we don't know either of the angles, so there are too many unknowns to use this equation. So the second equation requires us to have angles b and c and side lengths b and c. But again, while we know the angle of c, and we know the side length b, we don't know the measure of this angle b or the side length c, so we can't use this equation either. This last equation requires us to have the measure of the angle a and c and the side lengths a and c. And this time, we still don't have enough information to use this equation, so we should try the law of cosines. So in the law of cosines, we don't know c, but we do know a, b and the angle c, so we can use this equation. So we'll substitute in our values, which will give us the measure of the side c.