 We occasionally run into some problems when trying to solve a triangle. So let's try to solve a triangle with sides of lengths 7, 10, and 20. Well, let's do this the hard way and not draw a picture. Since we have no angles, we have to use the law of cosines. So we'll substitute in our values and solve for c. And at this point, one of two things will happen. If you have a cheap calculator, it will choke on trying to find an angle whose cosine is minus 251 over 140. On the other hand, if you have a good calculator, it will give you an angle that's absolutely meaningless. And here's the important thing to recognize. We tried to do this the hard way and not draw a picture. So we did a lot of work and came up with a meaningless answer. Instead, let's try to do this the easy way and draw a triangle where sides are lengths 7, 10, and 20. Now, for convenience, we'll make the longest side the base of the triangle. So we have this side length of 20 as our base and we'll make the other two sides of length 7 and 10. And if you look at your drawing, you realize there is a little bit of a problem. This indirect path has length 7 plus 10 equals 17, but that's shorter than this direct path, which has length 20. And what this means is that a triangle with lengths of 7, 10, and 20 can't exist. And so our solution is that no such triangle can exist. So what this says is that the side-side side case can be problematic. If you have all three sides of a triangle, then either the triangle can be solved or no such triangle exists. You can also solve a triangle if you have two angles on the common side, two angles in the non-common side, or two sides and the included angle. And in these cases, the triangle always exists and it's always unique. But what if you have two sides and the non-included angle? This gives you what's referred to as the side-side angle case. One way to look at it is the unknown third side makes an angle with the given side, so we'll just extend it and see if we can form a triangle. Several things may happen depending on the lengths of the sides. Now if the other known side is too short, we can't form a triangle. Or if the other side has just the right length, you may have a unique triangle, or you may have two distinct triangles. And what will determine this is whether or not you have 0, 1, or 2 solutions for the length of the third side. A useful idea to keep in mind in this case is that inside side angle, the far side swivels. So if possible, we'll solve a triangle with A equals 12, B equals 15, and the measure of angle A equals 40 degrees. So to begin with, we'll draw a picture. And the first important thing to notice is that we're in the side-side angle case, and since the far side swivels, we see we might get two solutions, or none. So to begin with, we can use the law of signs to find another angle. So we have angle A and sides A and B, so the law of signs that we want to use includes angle A and the sides A and B. Substituting in our known values. Then using our power tool gives us one angle. Now we have two angles, and we can find the third because the sum of the three angles must be 180 degrees. Now we have two sides and a bunch of angles, so we can use the law of cosines to find the third side, and we get one solution. But wait, there's another solution. When we found our first angle, we had to take the arc sign of a value. But our power tool only gave us one value for the arc sign, but there are more. Remember that if sine is theta is equal to z, then sine of 180 degrees minus theta will also equal z. So there is a second possibility for the measure of angle B, 180 degrees minus 53.46 degrees. Again, the measure of the three angles in a triangle must add up to 180 degrees. Since we've changed the measure of angle B, we will also change the measure of angle C. So we'll have to solve that again, which gives us a second possibility for the length of the side C. Or how about this? Since we want to make this problem difficult, we don't draw a picture. Well, since we'd actually like to get the correct answer, we'll draw a picture. Now we know the lengths of two sides, and we know the angle opposite side A has a measure of 40 degrees. So the third side will make a 40-degree angle, but we don't know how long it is. We have two sides and an angle, so we can choose one of the formulas in the law of sines. And since we have the sides A and B, and the angle A will choose this version of the formula, substituting in our values, and we can try to solve for B. And since sine of theta must be between minus 1 and 1, there is no angle for which sine of B is equal to 1.377. And so we know there is no triangle which has these sides and angles.