 In this video, consider the following oblique triangle with vertices ABC where the side length A is given as 2, the side length B is given as 6, and then the angle measurement of A is given as 30 degrees. I should mention that this is the side-side angle situation, the so-called ambiguous case. So we want to solve the remaining sides of this triangle. We notice because we have angle A and we have side-length A that we have an AOS, an angle opposite side pair. Whenever you have an AOS, this indicates to us we probably want to use the law of signs to help us solve for the remaining sides of this triangle. In particular, we're going to compare the AOS A with the AOS of B, which we have the side-length B but we don't have angle B. So using the law of signs, we're going to take sign of B over little B and that's going to equal sign of A over little A. And we want to solve for angle B right here. To do that, of course, you can times both sides by little B. So you get sign of B is equal to B over A times sign of A and then put in the information we have. B is 6, little A is 2, sign of 30 degrees. Now, sign of 30 degrees, that's, of course, 1 half. 2 goes into 6 exactly three times. Sign of 30 degrees is 1 half, like I said. And so we end up with sign of B is equal to 3 halves or 1.5 if you prefer the decimal there. This already might set off red flags for many of us here because, like, wait, sign of B equals 1.5. I thought you said earlier that sign is bounded between negative 1 and 1. Shouldn't it be true that sign of B is less than equal to 1 greater than negative 1? Yes, that's true. It doesn't matter what the angle is. The range of the sign function is between negative 1 and 1. Sign can never exceed 1 in its size. So it turns out this is not possible. Sign of B could not equal 1.5 here whatsoever. Now, if you didn't recognize that, immediately it's not a big deal. The next step you'd actually would do is you'd be trying to compute B equals sign inverse of 3 halves or 1.5 on your calculator. And then your calculator would give you some type of error of some kind, like D and E, like does not exist. It's undefined. Sign inverse of 3 halves is undefined. And so when you try to compute the sign ratio and you get something that's too big or too small, or your calculator tells you sign inverse doesn't exist for that value, that's an indicator that there is, in fact, no solution to solving this triangle. There is no triangle that satisfies the conditions that A is 2, B is 6, and angle A is 30 degrees. So this gives us an example of the ambiguous case that has no solution given side-side angle.