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Video: Proof of the Law of Sines

Lauren McNaughten

Work through a proof of the law of sines by finding equivalent expressions for the perpendicular height of a triangle using the sine ratio and equating and rearranging them into the format we are familiar with.

05:47

Video Transcript

In this video we’re going to look at one method of proving the law of sines. First of all, a reminder of what the law of sines is. So I have here a triangle, in which I’ve labelled the three angles as 𝐴, 𝐡, and 𝐢 using capital letters and then I’ve labelled the sides as π‘Ž, 𝑏, and 𝑐 using lowercase letters. And you can see that side π‘Ž is opposite angle 𝐴, side 𝑏 is opposite angle 𝐡, and side 𝑐 is opposite angle 𝐢.

The law of sines is written at the top in red here, and what it tells us is that the ratio between a side and the sine of its opposite angle is constant throughout the triangle.

So for any of these pairs, if I take the side length and divide it by sine of the opposite angle, then I get the same result, which is why I have π‘Ž over sin 𝐴 is equal to 𝑏 over sin 𝐡 and that’s also equal to 𝑐 over sin 𝐢.

We can also specify the law of sines in an alternative form where we invert each of these fractions, so we have sin 𝐴 over π‘Ž is equal to sin 𝐡 over 𝑏 and equal to sin 𝐢 over 𝑐, but it’s this version of the law of sines that we’re going to look at a proof for.

So the first stage in this proof, I’m going to draw in a perpendicular height from angle 𝐢 down to the base of this triangle 𝐴𝐡. So there’s this height here which I’ve labelled as β„Ž and you can see that it divides the triangle up into two smaller right-angled triangles which I’ve labelled as triangles one and two.

Now those triangles aren’t necessarily congruent to each other unless the original triangle was isosceles, so we’re assuming that they’re not the same as each other here. So as I’ve now got right-angled triangles, I can perform standard trigonometry using sine, cosine, or tangent.

And what I’m going to do is I’m gonna recall the definition of just the sine ratio in a right-angled triangle. Remember then that it’s defined like this: sine of an angle πœƒ is equal to the opposite divided by the hypotenuse.

So what I’m going to do then I’m going to work in triangle one first of all, and my first step is I’m going to label the three sides of this right-angled triangle in relation to angle 𝐴.

So I have the opposite, the adjacent, and the hypotenuse, and now I’m going to write down the sine ratio for angle 𝐴, so it’s opposite divided by hypotenuse, and that’s going to be β„Ž divided by lowercase 𝑏 using the notation in this diagram.

So I have sin 𝐴 is equal to β„Ž over 𝑏. I’m gonna do one rearrangement of this. I’m going to multiply both sides of this equation by 𝑏. And I’ve swapped the order of the two sides here, but it tells me that β„Ž is equal to 𝑏 multiplied by sin 𝐴, 𝑏 sin 𝐴.

I’m now going to move over to triangle two and do the same thing in relation to angle 𝐡. So that side β„Ž is still the opposite, but I’m going to fill in the hypotenuse and the adjacent as well. And now I’m going to write down the sine ratio for angle 𝐡. So opposite divided by hypotenuse, it will be β„Ž divided by π‘Ž.

I have then that sine of the angle 𝐡 is equal to β„Ž over π‘Ž. As I did before, I’m gonna do one rearrangement to this where I multiply both sides of this equation by π‘Ž. This tells me then that β„Ž is equal to π‘Ž sin 𝐡.

Now perhaps you can see what the next step’s going to be. I have got two expressions for β„Ž: one is 𝑏 sin 𝐴 and the other is π‘Ž sin 𝐡. And if they’re both equal to β„Ž, then I can set them equal to each other.

So I have then that 𝑏 sin 𝐴 is equal to π‘Ž sin 𝐡. Remember when I’m saying π‘Ž and 𝑏 here, there’s a difference between capital 𝐴 representing an angle and lowercase π‘Ž representing a side, and the same thing for 𝑏.

Now perhaps you can see this is looking close to the law of sines, but it isn’t quite there yet. In order to get this to the law of sines, I need to divide both sides of this equation by sin 𝐴 and by sin 𝐡.

If I just divide by sin 𝐴 first of all, then you’ll see that it cancels out the fact of sin 𝐴 on the left-hand side and appears in the denominator on the right. Now if I divide both sides by sin 𝐡, well then it cancels out the fact of sin 𝐡 in the numerator on the right and it now appears in the denominator on the left.

So what you see is that I have the first two parts of the law of sines. They’re written the opposite way round how they appear at the top, but I have that 𝑏 divided by sin 𝐡 is equal to π‘Ž divided by sin 𝐴.

Now that’s only demonstrated that two parts of this ratio are equal, but you could demonstrate it for the other pairs by drawing a perpendicular from 𝑏 to the opposite side for example.

So this gives you then a fairly concise proof of the law of sines. We started off with a triangle, drew in the perpendicular height from one of the vertices. We then used the sine ratio in each of the right-angled triangles in turn to get an expression for this perpendicular height β„Ž, equated the two expressions, and then with a little bit of rearranging we had our proof of the law of sines.