Hello wonderful mathematics people. I'm Anna Cox from Kella Community College. Law of Science. Law of science is going to be used to find unknown angles and or sides in a triangle that is not a right triangle. An oblique triangle is a triangle that is not a right triangle. If we look at any triangle and we labeled the angles and the opposite sides accordingly, sides are labeled with little lowercase and angles are labeled with capital. If I drop to height from one of the vertex, let's call it H. If I look at this now, I do have two right triangles. So if we look at the triangle on the left, we'd have sine of A equaling H / B, or H would equal B sine A. If we look at the triangle on the right, we'd have sine B equaling H / A or a sine B equaling H If H = 2 different things, by the transitive property, those two things have to be equal. So B sine A has to equal a sine B. Now depending on what we're trying to find depends on how we write this. If we're trying to find an unknown side, we would divide each side by sine A sine B. So if we have B sine A and we divide it by sine A sine B, the sine A's will cancel. And if we take the other side a sine B and divide it by sine A sine B, the sine B's will cancel. So we would come up with B over sine B equaling A over sine A. If we were trying to find an unknown angle. Taking that same expression, we would divide each side by AB. If I went back here and I quickly erased this and divided by AB and AB, the B's at cancel and we'd end up with sine A over side A equaling sine B over little B or side B. Now we could drop a vertices from or an altitude from a different vertices to show that that's true. Also for side C, we're going to have a couple different cases for this. We're going to have a case where we might have two triangles. So if we know a side, say we know side B and we know angle A, then we know side A. So if side A was something pretty big, say this length here, if we thought about it, that we could draw as a circle, as a radius, so that everything on the circle would be side A, length A. Well, that would also intersect over here. So visually we'd have two different triangles. We'd have length A here, we'd have length A here. So this would be two triangles. We could also have one triangle. So if we know side B, we know angle A, and we know side A, side A might be something fairly large. If it was something large, when we draw in our circle, now we're only going to intersect that line. Wow, that's a bad circle. Sorry, we're only going to intersect that line one time. For one triangle. This would be the center of my circle. I realize it doesn't look like it at all, but the idea is there. If it had no triangle, side A would be so short that it wouldn't intersect the length at all. So if we look at this, say our length, A is this distance from the center and hence you can see this one would be no triangle because it never intersected. So let's look at a couple examples and it all depends on what we're given. So on this first example, we're actually given our two angles. So our third angle here is going to be 180° -, 45° -, 25°. So if we grab our handy dandy calculators 180 -, 44 -, 25 or perhaps we don't need our calculator, we're going to get 111 degrees to find side B&C. If we think about drawing our triangle, we know that A was 44 and B is 25. As a rule of thumb, we always want to use the Givens because they're exacts. So side A is 12. So to get side B over here, we're going to say B over sine of 25 is going to equal 12 over sine of 44. Now when you're doing this, make sure that you're in in degree mode and if you're not, you're going to want to change to degree mode to find side B. We're going to then cross multiply and get 12 sine 25 divided by sine 44, so 12 sine 25 divided by sine 44, which will give us about 7.30. Frequently it's 2 decimal places unless specified otherwise. Or even a better rule of thumb is if it gives it to you in whole numbers, you should give your answer back in whole numbers. We're going to do 2 decimal places for now because of rounding errors. If we went to whole numbers, we are really compounding possible rounding errors. So to get our last side, our side C, we'd say C over sign 111 because we found 111 degrees a minute ago. Once again, I'm going to use what's given because that was the exact with no rounding error. SO12 over sine of 44. So to get C, that's just 12 sine 111 degrees over sine 44°. So 12 sine 111 divided by sine 44 is going to give me AC that's 16.13. And that's the first example. Looking at this next example, we know her angle A is 37. We know our side A is 12. Our side A is opposite angle A. We know our side B is 16.1 and it's opposite angle B We don't know our side C and then our and the angle C So we're going to start by saying angle B or sine of B / 16.1 equal sine of 37 / 12. So sine B is going to equal 16.1 sine 37° / 12. So B is going to be the sine inverse of 16.1 sine 37° / 12. When we put this into our calculator, we're going to get 53.85°. Now sine if we think about our unit circle is actually positive in the 1st and 2nd quadrants. So we might have two triangles because if 53.85 is here, we know that there's another angle value with the same output or the same Y the same height in our unit circle. So if by reference angles this is 53.85 also, we have to take into account that this angle here might also work. So we're always going to find a second angle, call it B2, and let's call the other one B1. And B2 is always going to be 180 -, B one. So if we have 180 and we -53.85, we're going to get 126.15. Now the angle C, because the angles all add up to 180, we're going to say 180 minus our given angle of 37 -, 53.85 South, 180 -, 37 -, 53.85, that's going to give us 89.15. So there's definitely 1 triangle. Let's see if there's a second triangle. We're going to take 180 minus our given angle, 37 minus our B2, which was 126.15 S, 180 -, 37 -, 126.15. That's going to give us 16.85 S. This is going to have two triangles. We're going to have AB1 and AB2. We're going to have AC1 and AC2, which means that we have to find side C1 and side C2. So we're going to say 12 over sine 37 is going to equal side C over our C1. Sine 89.15 SC is going to equal 12 sine 89.15 divided by sine 37. So 12 sine 89.15 divided by sine 37, which will give us 19.94 and that's actually our C1. To get our C2, we have to use our second C angle. So to get the C2 it's going to be 12 over sine 37. Again, that part doesn't change our C2 over sine of our second angle or 16.85. So when we cross multiply there, we're going to have 12 times 16.85 divided by sine of 37 and that's going to give us AC2 value of 5.78. Let's look at another example. On this next example, we know our A and our we know our side A and our angle A and our side C So we're going to have sine of C / 8.9 equaling sine of 63° / 10. So C is going to equal sine inverse of 8.9 times sine of 63. That's just cross multiplying divided by 10. So then when we put that into our calculator, make sure once again that you're in the correct degrees. We're going to get C1 being 52.47. Now let's realize that we could have AC2. If we have AC2, it's going to be 180 minus that 52.47. So 180 -, 52.47 is going to give us 127.53°. So, so far we have two triangles. Now let's look at our BS. Our B1 is going to be 180°, a full triangle minus our two known angles. We know 63 and we just found 52.47, so 180 -, 63 -, 52.47, 64.53. So we definitely have one triangle. Let's see if we have two. Our B2 would be 180 minus the two known angles. 63 was known and we just found 127.53, so 180 -, 63 -, 127.53. That's going to give us -10 and then right? Or in triangles, we're not going to have negative, so no second triangle here. There's only one triangle, so we need to find side B for the one triangle. So we're going to have B over sine 64.53 equaling going back to the original. We want to use the Givens because they're exact, they're not approximates. So 10 over sine 63. So B is going to equal 10 sine 64.53 divided by sine 63 or 10.13. So that's an example of when we only had one triangle. We didn't have a second triangle because we got a negative angle when we were finding the third angle. Let's look at one more example here. We're going to find sine C over 37 and that's going to equal sine of 112 / 42.1. So we know C is going to equal sine inverse of 37 sine 112 divided by 42.1. So when we put that into our calculators, we get 54 .57 as our C1 RC 280 -, 54 point 57125.43. RB one is 180 minus our 112 -, 54.57, so 180 - 112 - 54 point 5713.43. So we know there's one triangle. RB2 is the 180 -, 112 -, 125.43. And that's a -57.43 can't have a negative angle and a triangle. So we're only going to have one triangle here. So now we're going to find side B. So B over sine 13.43 is going to equal. Always use the Givens. They're more exact for 2.1 over sine 112. So 42.1 times sine of 13.43 divided by sine of 112 should give us B being 10.55. Let's do one more example. Let's do if capital A is 75° little, A is 51 and B is 71. So we're going to use our law signs. So we're going to say sine of B / 71 equal sine of 75 / 51. So B equals sine inverse of 71, sine 75 all over 51. So sine inverse 71, sine 75° / 51. When I put this into my calculator, we get an undefined domain error. If we get an undefined domain error, it means that we can't find the sine inverse of this number. This number in here must not be between -1 and one because we know sine inverse has to be in between -1 and one. Hence this is an example of a no triangle. Thank you and have a wonderful day. This is Anna Cox.