 The trigonometric functions and the laws of sine and cosine allow us to do what's known as solving a triangle Solving a triangle is based on a couple of important ideas from geometry. A triangle may be uniquely determined if we know all three sides. This is sometimes referred to as side, side, side, SSS, two angles and the common side, angle, side, angle, two angles and a side, angle, angle, side, and two sides and the included angle, side, angle, side. We say that the triangle is solved once we know the lengths of all three sides and the measures of all three angles. So let's take a triangle like this where the measure of A is 47 degrees, the measure of B is 17, and the length of the side AB is 8 units. Let's solve the triangle. In other words, we want to find the measure of all three angles and the lengths of all three sides. Well, let's draw a picture. Now the picture is not part of the portfolio. We're going to use to enter art school. Thank goodness. But rather it's a way of organizing all of our information. And since this is triangle ABC, we should probably label the vertices ABC, then set down the angles and lengths that we know. Angle A is 47 degrees, angle B is 17 degrees, and the length of side AB is 8 units. It will also be convenient to remember that we can label these sides by the angle they're across from. So this side here we'll call little A, this side will be little B, and this side will be little C. And remember, we're trying to find all three angles and all three sides, and we might start by noting that we have the angle A, B, and the length C. So based on this, we see that we need to know two side lengths and one angle. So how do we find them? Well, to begin with, one useful theorem is that the sums of the measures of the three angles in a triangle is 180 degrees, which means that if we know two of the angles, and we do, we can find the third angle. And so let's set that up. We know that the sum of the measures is 180 degrees. We know the measure of angle A and the measure of angle B, so let's substitute that in and solve for the measure of angle C. And if it's not written down, it didn't happen. How about the lengths of the sides A and B? For that, we have two theorems that relate side lengths and angles, the law of sines, and the law of cosines. So the law of cosines requires us to know two sides and an angle, but we only know one side. And so we see if we can use the law of sines. Since we have the measure of angle C and the side length C and the measure of angle A, we can find the side length of A using the law of sines. So substituting our values in. Now, while we could find sine of 116 and sine of 47 degrees, let's solve for A first so we don't have to carry around a bunch of messy decimals. So solving for A gives us, and now we can find sine of 47 degrees and sine of 116 degrees. And if it's not written down, it didn't happen. And now we want to find the length of B. Now, for using the law of sines, we have to use this angle 17 degrees, but we have two choices. We can either use this length C equals 8 and angle C equals 116 degrees, or this side length A about 6.5096 and the angle A 47 degrees. To decide which to use, a useful idea to keep in mind is use the numbers that you trust the most. In this case, this side length C equals 8. We have to trust because we're given that as part of the problem. If we don't trust this number, we can't solve the problem. So we'll want to use this side length C and this angle 116 degrees to set up our formula for the law of sines. Solving for B and computing gives us our third side length. And now we have all three angles and all three sides, and we have solved the triangle. Or have we? Well, probably if we did all of our computations correct. And here it's useful to do at least one check, which is based on the following theorem. The longest side in a triangle is opposite the largest angle, and the smallest is opposite the smallest angle. So if we take a look at our three angles, the largest angle is C, and the side length is 8, and we verify that this is actually the longest side of the triangle. Likewise, the smallest angle is B, and the side across from it is also the shortest side of the triangle. While this is no guarantee we've done all of our computations perfectly correctly, it tends to catch very serious errors like not having your calculator in the proper mode. Another one of the UD cases is where we have all three sides of the triangle. So for example, maybe we want to solve a triangle with sides of length 8, 9, and 13. So let's go ahead and sketch a triangle where the sides have length 8, 9, and 13. Now it doesn't really matter which vertex we call A, B, and C, and we should always indicate the side opposite with the lowercase version of the vertex. To determine how to proceed, we can pick any formula and see if we can use it. So what formulas do we know? Well, we know the sum of the angles in a triangle, we know the three equations that correspond to the law of sines, we know the Pythagorean theorem, and we know the law of cosines. First off, because this is not guaranteed to be a right triangle, we can't use the Pythagorean theorem. Next, we don't know any of the angles in this triangle, so this sum of the angles at 180 degrees isn't useful. There are too many unknowns in this equation. How about our law of sines equations? This first equation requires us to know the measures of the angles A and B, as well as the side lengths A and B. And since we don't know the measures of any angle, we have two unknowns in this equation, and we can't solve it. The same argument applies for the second equation. We don't know the measures of angle B and C, and so there are too many unknowns in this equation. On the other hand, in this equation, oh wait, we still don't know the measure of angle A and C, what about this equation? Well here, we know the lengths of all three sides, A, B, and C, so the only unknown in this equation is the measure of this angle C, and that means this is the equation we want to start with. So we have A, B, and C, we can use the law of cosines. Substituting those values in, then solving for C gives us, well now we have two sides at an angle, which means we can use the law of sines. So we'll use our first equation from the law of sines, substitute in our values, and again we get down to the point where the sine of some angle is approximately some numerical value. So again, we have to use the inverse sine function to find our angle is about 36.7988 degrees. Now we could do the same thing to find this measure of angle B, but we might as well use our simpler relationship, the sum of the three angles is 180 degrees, and we know two of the angles already. So we'll substitute those values in, and solve for B. And it's worth doing a quick check. We want to make sure that the longest side is across from the largest angle, and the shortest side should be across from the smallest angle.