Let's say side A is 10 and side B is 20 and angle C is 60 degrees. Go ahead and solve the triangle. So first let's draw it.
So this is angle A, B, and C. So angle C is 60 degrees, side A is 10, side B is 20. So what we have is a side, angle, side triangle. Can we use law of sines to solve the triangle? In order to use law of sines, you need to have two of the same letter. Notice that we can't use it.
We have one of each different letter. So in this case, we need to use the law of cosines. If you try to use the law of sines, you're going to miss something. For example, let's say if we try to use a over sine a, which is equal to b over sine...
B. It's not going to work. We have A and B, but we don't have angle A, nor do we know angle B.
And if we try to use B over sine B, which is equal to C over sine C, we're still missing angle B, and we're missing sine C. So whenever you have all different letters, you cannot use the Law of Sines to solve it. However, there's something else that we can use, and that is the Law of Cosines. So here's the formula that you need. c squared is equal to a squared plus b squared minus 2ab cosine of angle c.
Now you can change it up and write two other forms. A squared is equal to B squared plus C squared minus 2BC cosine of angle A. Or, B squared is equal to A squared plus C squared minus 2AC cosine of angle B. You can use any one of these three forms, but I'm going to use the first one because we have everything to use that formula.
We have side A and B. A is 10, B is 20, and we have angle C, which is 60 degrees. So we can use this to find side C. 10 squared is 100, 20 times 20 is 400, and 2 times 10 is 20, times another 20, that's 400 as well.
Now, cosine 60 is 1 half. 100 plus 400 is 500, and half of 400 is 200. So, c squared is equal to 300. So therefore, C is the square root of 300, which is 10 root 3, or 17.32. So now that we have side C, we could use the law of sines to figure out everything else.
So let's use the law of sines to find angle B. C over sine C is equal to B over sine B. Side C is 17.32 and angle C is 30. B is 20. Let's go ahead and find angle B.
So let's cross multiply. So 20 times sine 60, that's 17.32. And that's equal to 17.32 sine b. So if we divide both sides by 17.32, what that means is that sine b is equal to 1. So therefore, B is the arc sine of 1, which is 90 degrees. So there's only one answer here, because if you do 180 minus 90, you're going to get 90 again.
Now to find angle A, that's going to be 180 minus 90 minus 60, which is going to be 30. And so that's how you can solve this particular triangle. It turns out that it's a right triangle. Let's say that side A is 7, side B is 8, side C is 9. Use the law of cosines to solve the triangle.
So this time, we have all three sides. So what we have is a side-side-side triangle. So let's start with this formula. C squared is equal to a squared plus b squared minus 2ab cosine of angle c.
So first, let's subtract both sides by a squared and b squared. If we move it to the left... left side we're going to have c squared minus a squared minus b squared is equal to negative 2ab cosine of c.
Now let's divide both sides by negative 2ab. So this is the formula we're going to use. Cosine of angle C is equal to C squared minus A squared minus B squared divided by negative 2AB.
C is 9. A is 7. And B is 8. 9 squared is 81, 7 squared is 49, and 8 squared is 64. So 81 minus 49 minus 64, that's equal to negative 32. And then 2 times 7 times 8, that's 112. So negative 32 divided by negative 112, that's 2 over 7, which as a decimal is 0.2857. That's equal to cosine of angle C. So angle C is going to be the arc cosine of that number, 0.2857. So you should get about 73.3 degrees, actually 0.4 for angle C. Now let's use the law of cosines to figure out everything else.
So let's start with this equation. C over sine C is equal to A over sine A. So C is 9. Angle C is 73.4. Side A is 7. Now let's cross multiply. 7 times sine 73.4, that's about 6.708.
And that's equal to 9 times sine 8. So 6.708 divided by 9, that's about 0.7453. So angle A is going to be the arc sine of that number. So that's 48.2 degrees. Now, let's find the other answer. So, A could be 180 minus 48.2, which is 131.8.
But this answer is not possible, because if we add this to the pre-existent angle, that exceeds 180. So, therefore, there's only one possible solution, one triangle that can be formed. So, A, let's write the answer, that's 48.2 degrees. And now let's calculate angle B. So B is going to be 180 minus 73.4 minus 48.2, and that's 58.4 degrees. So if you have all three sides, you need to use the law of cosines to find the first angle.
And then you can use the law of sines to find everything else. So that's it for this lesson.