### Video Transcript

In this video, weโre gonna look at how to use the sine rule to answer questions in which there is more than one possible interpretation of the question given the information. For example, if a triangle ๐ด๐ต๐ถ is described as having side ๐ด๐ต with length five centimetres and side ๐ต๐ถ with length four centimetres, and the measure or size of angle ๐ต๐ด๐ถ is forty-five degrees, then there are two possible situations. Side ๐ต๐ถ could be in two different orientations and each gives quite a different answer to the question: Find the measure of angle ๐ด๐ถ๐ต.

Now first remember that the sine rule states in any given triangle, the ratio of the length of a side and the sign of its opposite angle is the same for any side. So if we have triangle with vertices ๐ด, ๐ต, and ๐ถ and we label the sides like this, then the ratio ๐ over sin ๐ด gives the same result as ๐ over sin ๐ต, which also gives the same result as ๐ over sin ๐ถ. Although sometimes itโs more convenient to put those ratios the other way up, so sin ๐ด over ๐ is equal to sin ๐ต over ๐ is equal to sin ๐ถ over ๐. Now weโre gonna use this sine rule in this video to answer some ambiguous questions.

Now going back to the question we started looking at, we can see the side length ๐ is four centimetres and side length ๐ is five centimetres. And also the measure or size of angle ๐ต๐ด๐ถ is forty-five degrees. Now given that the angle weโre looking for is angle ๐ถ there, we can use this part of the sine rule: sin of angle ๐ด over ๐ is equal to sin of angle ๐ถ over ๐. So sin forty-five over four is equal to sin ๐ถ over five. Well I can rearrange this by multiplying both sides by five. So thatโs sin ๐ถ is equal to five times sin forty-five over four. And five sin forty-five over four is five root two over eight. So ๐ is the inverse sin of five root two over eight, and my calculator tells me that that, to two decimal places, is sixty-two point one one degrees, so thereโs our answer, well or is it? Thatโs one of our answers. Remember we said there were two possible answers. If we think back to our original scenarios, it looks like weโve solved this question here, so weโve still got to think about what would the angle ๐ต if side ๐ต๐ถ came in back towards ๐ rather than out away from ๐. Now thinking back to a unit circle, sin of ๐ here is equal to the length of the opposite side in that triangle divided by the length of the hypotenuse.

Well the length of the opposite side is the height of that triangle and the hypotenuse is one, so the sin ๐ is the height divided by one. So sin ๐ is the height of that triangle. And in this case, that is a positive amount here. But thereโs an equivalent triangle over here. So if I have an angle of ๐ the same here, they will have the same height. So in other words, there are two values of ๐ between zero and a hundred and eighty degrees which give triangles of that particular height which have the same sine value. And they are gonna have angle sizes of ๐ and whatever this measure is here. So itโs a hundred and eighty minus ๐.

Now in our case, we said ๐ was sixty-two point one one degrees. So in our case, this other angle here is gonna be a hundred and eighty minus sixty-two point one one degrees, which is a hundred and seventeen point eight nine degrees. So now we found our two possible answers: sixty-two point one one degrees or one hundred and seventeen point eight nine degrees. So the information we were given in the first place was ambiguous and it lead to two possible scenarios. So in the first case, we could have had an angle there of sixty-two point one one degrees and in the second case there it could have been a hundred and seventeen point eight nine degrees.

Okay letโs have a look at another question. In triangle ๐ด๐ต๐ถ, the measure of angle ๐ด, in other words, the size of angle ๐ด is seventy-two degrees. The length ๐ด๐ถ is fifty centimetres and the length ๐ต๐ถ is ten centimetres. If such a triangle exists, find the measure of angle ๐ต correct to two decimal places. Okay well thereโs a bit of a clue in this question. It says if such a triangle exists, find the measure of an angle ๐ต correct to two decimal places. So first of all, we have to check if itโs possible for that triangle to exist.

So first, letโs do a sketch. Now it doesnโt matter that this is dramatically not to scale. We can still do our calculations based on what weโve written down. So weโve just written in the side names here, weโve got ๐, little ๐, is ten centimetres and little side ๐ is fifty centimetres. Now weโve got the measure of angle ๐ด and the length of side ๐. Weโve got the length of side ๐. And although we havenโt got the measure of angle ๐ต there, but we can use the sine rule on angles ๐ด and ๐ต.

And remember that means that the sine of angle ๐ต divided by the length of side ๐ is equal to the sine of angle ๐ด divided by the length of side ๐. So that means that the sin of angle ๐ต divided by fifty is equal to the sin of seventy-two degrees divided by ten. Now if I multiply both sides by fifty, the fifty is cancelled on the left-hand side, so that leaves me with sin ๐ต is equal to fifty sin seventy-two divided by ten. And my calculator tells me that fifty sin seventy-two over ten is equal to four point seven five five two eight two five eight one and so on. But the sine of an angle must be between negative one and positive one, so this is impossible. Such a triangle cannot possibly exist.

If we try to work out sine to the minus one of all this lot, our calculator would give us an error. The numbers we were given in the question would lead to sine of an angle giving an answer of over four which is impossible. Sine of an angle must always be negative one and one, so such a triangle cannot exist. Thatโs our answer.

Letโs look at one more question then. In triangle ๐ด๐ต๐ถ, the measure of angle ๐ด is forty-eight degrees. Side ๐ is nineteen centimetres. Side ๐ is twenty-one centimetres. If such a triangle exists, find the possible measures of the other angles and the possible lengths of side ๐, all correct to one decimal place. So again, letโs start off by doing a sketch. And again, we havenโt done this to scale, so it doesnโt necessarily that accurate. Now we know the size of angle ๐ด and the size of side ๐. We know the size of side ๐, but we donโt know the size of angle ๐ถ.

So we can use this form of the sine rule: sin ๐ถ over ๐ is equal to sin ๐ด over ๐. So filling in those unknown values, sin ๐ถ over twenty-one is equal to sin forty-eight over nineteen. Now we can multiply both sides by twenty-one. So the twenty-one cancels on the left-hand side and we get sin ๐ถ is equal to twenty-one sin forty-eight over nineteen. Now that means that ๐ is equal to the inverse sine of all this lot. Iโm gonna put that into the calculator. This is what comes out: fifty-five point two two two two three two zero one. Weโve only been asked for one decimal place. So weโve got ๐ถ is equal to fifty-five point two degrees. Thatโs our first answer.

But remember, this side ๐ต๐ถ could have been orientated like that or it could have been oriented like that, so thereโs another possible value here. We found this answer here, but thereโs also another version of the answer here. So again, thinking back to our unit circle, we found this answer here. But thereโs another solution that gives the same height of the triangle, the same sin ๐ value between zero and a hundred and eighty and itโs this angle here. So thatโs a hundred and eighty minus fifty-five point two. So that gives us two possible different values for ๐.

Now for the rest of the question, weโve got to be quite careful to keep the right combination of values together. You canโt just mix and match these throughout the rest of question. Youโve got to keep the right angles with the right angles. Now if ๐ถ is equal to fifty-five point two degrees, then angle ๐ต here is equal to a hundred and eighty minus forty-eight minus fifty-five point two, because angles in a triangle sum to a hundred and eighty degrees. And that works out to be seventy-six point eight degrees. But if ๐ถ was a hundred and twenty-four point eight degrees, then ๐ต is gonna be a hundred and eighty minus this forty-eight minus the hundred and twenty-four point eight. And that would give us an answer for ๐ต of seven point two degrees. So weโve got two different versions of the triangle with two different corresponding values of ๐ต and ๐ถ.

Now to get a nice accurate answer for the length of side ๐ and round to one decimal place is actually quite tricky. We could use the cosine rule or we could use the sine rule with two possible different values for ๐. Well letโs stick to the sine rule for now, but weโll keep a few extra places of accuracy in our calculation for angle ๐ต. Now remember in our calculator, we had all these decimal places. Now we donโt need to keep all of these, we-we will do in this particular question just to keep the most accuracy that we can. And weโve said that ๐ถ was fifty-five point two two two two three two o one and so on. And if we use that figure for calculating the measures of angle ๐ต, weโd have get-weโd have got seventy-six point seven seven seven seven six seven nine nine and seven point two two two two three two zero one. So theyโre the values we gonna use in the next part of the calculation.

Then going back to our sine rule, ๐ over sin ๐ต equals ๐ over sin ๐ด. Now these values, ๐ and sin ๐ด, I was given in the question itself, so these are as accurate as they can possibly be. And weโre using the most accurate version of the answer we can for sin ๐ต. So when I work this out, Iโve got ๐ over sin seventy-six point all that lot is equal to nineteen over sin forty-eight. So if I multiply both sides by sin seventy-six point all that lot, I get nineteen sin seventy-six point all that lot over sin forty-eight, which my calculator tells me to one decimal place is twenty-four point nine centimetres. So when ๐ถ is fifty-five point two degrees and ๐ต is seventy-six point eight degrees, then the length of side ๐ is twenty-four point nine centimetres. So just quickly letโs do the calculation again with the other versions.

So weโre using the same values for the length of side ๐ and the measure of angle ๐ด, but now weโre using a size of seven point two two two two three two o one for angle ๐ต. So ๐ over sin seven point two that lot is equal to nineteen over sin forty-eight. And when we rearrange that, we get ๐ต is equal to three point two to one decimal place. So when ๐ถ is a hundred and twenty-four point eight degrees and ๐ต is seven point two degrees, then to one centimetre the length of side ๐ would be three point two centimetres.

So the last thing that remains is for us to make it absolutely clear which measurements go with which measurements. And so weโre writing out our answer like this when ๐ถ is fifty-five point two degrees, then ๐ต is seventy-six point eight degrees. And the length of side ๐ is twenty-four point nine centimetres all correct to one decimal place. And when ๐ถ is a one hundred twenty-four point eight degrees, then ๐ต would only be seven point two degrees, and the length of side ๐ would only be three point two centimetres. And itโs worth just writing in the level of accuracy that youโve given your answers to.