Sine and cosine rules

In this video, we will look at some trigonometric methods involving using the sine and cosine rules with Dr. Ria Symonds.

The learning objectives for this video are to be able to use the sine and the cosine rule to find a missing angle or a missing side of a triangle, and also to use an alternate formulae in order to find the area of a triangle. Let’s begin our video by looking at a triangle and how we may label this to remember the cosine and the sine rules.

So here is my triangle. And inside, I have labelled my interior angles large A, large B, and large C. And opposite each angle is a side, and I have labelled the side with the same letter. So opposite the angle large A is a side small a, and the same for b and c. Now, if we label our triangles in this way, it will help us remember the sine and cosine rules. Let’s begin with the cosine rule and decide when we might use that as a method.

So the cosine is normally used when we know either three sides of a triangle. So perhaps we know A, B, and C. Or we only know two sides of our triangle– perhaps here B and C– and also the included angle, or inclusive angle. So here I know the angle between the two sides. Now, there are two formulaes to remember. You can remember one of them and rearrange the other. So one way of remembering the cosine rule is via the angle. So cosine of a is equal to b squared plus c squared minus a squared over 2bc. And if you had alternate sides and the alternate angle, you would just swap the letters around.

Otherwise, you could rearrange that formula, perhaps here, to find a squared. So if we want to find the side a– so over here– then we could alternatively remember the cosine rule as follows– a squared equals b squared plus c squared minus 2bc cos A. Now, there is an alternate rule we could use called the sine rule. And this time, we use the sine rule if we have two sides and an angle where the angle is not necessarily included, or two angles and a side. So we need three bits of information. And that’s because the sine rule looks like this. It’s the ratios of the sine of the angle over the side itself. So perhaps we know these pieces of information.

Maybe we want to find the angle a again, but this time, we’ll be finding sine of the angle. And we know the opposite side a, and we also know this angle B and the opposite side b. So we need to make sure we’ve got information relating to the same sides and angles. Alternatively, maybe that we have these pieces of information over here, or we could use the a and the c parts together. And the final formulae we’ll look at today is the formulae for finding the area of a triangle. This is given as 1/2ab sine c.

So again, if you label your triangle in this way, that should help you decide what are the lengths a and b and the angle C. OK, let’s look at a couple of examples of how to use those formulae. So in my first example, I have two sides and an angle. And notice that the angle here is not included. So that tells me already I’ll probably need the sign rather than the cosine rule. Now for us to find two parts of information, the angle CAB, the angle CAB can be recognised by the letter in the middle here. So I’m looking at the angle A. And I also want the length large AB. So the length AB this side here.

And if we used our notation to label the triangle, that’s small c. So here’s small b and here’s small a. So let’s begin by finding that angle first, CAB. So to find the angle, I’m going to use the sine rule, and I’m going to use the sine rule using sine of a over a equals sine of b over b. And that’s because I know the side a, the side b, and I know sine of b. So I could find the sine of this angle over here. So let’s put in all of my bits of information. So sine a over side A, which is 10, is equal to sine of b, which is 63 degrees over b, which is 14.

But let’s rearrange that to find sine of a. So that’s 10 lots of sine of 63 degrees over 14. And you can put that into your calculator to find a decimal answer. Now make sure you keep as many decimal places as possible. And also make sure that your calculator is in degrees since the angle is given in degrees. And then we’ll want to find inverse sine of whatever that value is from your calculator, which I get to be something along these lines here 0.6364 dot, dot, dot. So, therefore I find that A as an angle is equal to 39.5 degrees to one decimal place.

And you can give as many decimal places as you like as long as you’ve rounded correctly. OK, now we’ve found that angle, let’s find the length AB. So the length AB, as I said, is given here, and it’s our small c. We also know that this angle down here now is 39.5 degrees to one decimal place. Now we can use the sine formulae again or the sign rule again. But I’m going to need to know what this angle is here because alternatively, I need to find the side small c. So really to use the sine rule, I need the angle large C.

And I can find that because I know that the angles of a triangle add up to 180 degrees. So if I subtract those two other angles like so, that will help me find the angle C. Now remember we rounded our angle 39.5 up. So I’m just going to go back to my calculator to find what the decimal was. And you’ll find that if you take the full decimal away from 180 degrees as well as the 63, that we should find it is in fact 77.47365 dot, dot, dot. Try and keep as many decimal places as you can. We can use the sine rule. So sine of c over c equals sine of b over b.

Let’s put in all our bits of information.

We need to find the side small c. And again, we can rearrange. So this times, C is going to equal sine of 77.47 times 14 over sine of 63. Put all of this into your calculator. You should once again find that you get an answer something along these lines. So I’ve given it to one decimal place, 15.3 centimetres. And that’s the answer to our first example. Let’s try a second example now. So in this example, I have a slightly different angle. I know two sides, and I know an inclusive angle. To find an inclusive angle, I’m probably going to use the cosine rule. Let’s label a power triangle also. So I’ve got small x, small z, small y.

Now, the first thing we want to do is calculate the area of the triangle, XYZ. And since I know two sides and an angle, I can use the new formulae we’ve seen. So we’re looking for the area. And in terms of the sides of our triangle, we’ve got a 1/2 times the two sides, y and z, times the inclusive angle, which is sine of 79 degrees. So y and z are 10 and 8. And our angle, as I’ve already put, is 79. We’ll just change that to an x so we know exactly where the formula is coming from. Once again, put this into your calculator. Make sure that you are in degrees.

And we should find that the area of this triangle is 39.27 centimetres squared. So this time, I’m using two decimal places, and our units are centimetres squared. Finally, let’s do the length of YZ. So to find the length, we want this side here. This is YZ, so small x. So this time, I’m going to use the cosine rule. So the cosine rule is given as x squared equals the other two sides added together squared– so that’s y squared plus z squared– minus 2 lots of yz times cosine of the angle, the opposite angle now, which is x, or the inclusive angle. So let’s put those values in. So here’s Z. Here’s Y.

So I’ve got 10 squared plus 8 squared minus 2 lots of 10 times 8 times cosine of 79. Again, we just plug the values into our calculator. We’ll find that x squared is equal to 133.47 with lots of decimal places. So to find x, we need to square root that value. It’s likely to be 100. So x is equal to the square root of 133 dot, dot, dot, dot. And that equals 11.55 centimetres, once again, to two decimal places. So that was two examples to see how we use the cosine, the sine, rule and also the area of a triangle. So let’s check that we have indeed hit our learning objectives.

So we’ve used the sine and the cosine rule to find missing angles all sides. And we’ve also used that alternate formulae defined in the area of a triangle. Now, go back, try some more questions. Ensure that you fully understand the methods that we’ve seen today and then move on to your next topic.