Where do Sin, Cos and Tan Actually Come From - Origins of Trigonometry - Part 1

Depending on where you are with your studies, chances are you've seen or heard of sine, cos and tan. They are found as buttons on pretty much every scientific calculator in existence. They are incredibly important and useful in the world of mathematics. People often ask where do they actually come from? I remember asking my teacher the same question in year 9 and that's what we'll be looking to find out in this video. To give us some idea of how they've come about, consider how pi is seen with circles. A long time ago, someone thought it would be a good idea to measure a circle's circumference, which is distance around the outside, and divide it by its diameter. Doing this for this circle, we get 3.141. Doing it again for this circle, we get 3.141. And the same again for this circle. And this will be true for every circle in the universe. This number that we always get is called pi and that is how we came up with the formula to work out the circumference of a circle, pi times diameter. It turns out we can do something similar with right angle triangles. Now, let's look at this right angle triangle. The angle here is 60 degrees. I've measured sides which have lengths of 2, 1 and 1.732 and we can just assume it's centimetres. It's always a good idea to label the sides. I'm going to call this side H for hypotenuse because it's the longest side, this side O because it's opposite the angle 60, and this side A because it's adjacent to the angle. I'm going to do the opposite side divided by the hypotenuse just to see what I get. So it's 1.732 divided by 2 which is equal to 0.866. I'll just make a note of that and move on. I've got another larger right angle triangle now. The angle here is still 60 degrees but I'm going to do the same thing. Label the sides as before and do the opposite side divided by the hypotenuse. So 5.196 divided by 6 and that's equal to 0.866 again. Interesting, different triangle but the same exact answer. Again, I've got a smaller right angle triangle now still with an angle of 60 degrees. Doing the same as before, dividing the opposite side by the hypotenuse, I get 0.866 again. And yes, it turns out that for any right angle triangle in the universe with a 60 degree angle, if you measure the opposite side and divide it by the hypotenuse, you will always get 0.866. You can even try this yourself. Draw any right angle triangle with one of the other angles as 60 degrees. Then measure the side opposite the 60 degree angle and divide it by the hypotenuse and you will always get 0.866. It looks like I've just discovered something. So let me make this into an official theorem. And I can use my theorem to find missing sides when one of them is missing. I've got a right angle triangle here. The longest side they've given, which is 5 kilometres, but the side labelled x is missing. Let me label them like before. Well, I know from my theorem that the opposite side, which is x, divided by the hypotenuse 5 will be 0.866. And I can just use algebra or work backwards to find x. Something divided by 5 is 0.866. So it's 0.866 times 5, which is 4.33 kilometres. Going back to my theorem, I'm going to keep a record of this in a table because it will make my life easier when I need to find missing sides in the future. And I'll also plot my results on a graph because we like graphs and they help us notice patterns and such. So far, all the right angle triangles we've looked at had the other angle as 60 degrees. But as we know, those other angles in a right angle triangle won't always be 60. In this triangle with 60 degrees that we looked at already, let's look at the other angle. Well, angles in a triangle add up to 180. So I can just subtract the 60 and the 90 from 180 to give me 30 degrees. I'll label the sides as before. This time, the opposite side is opposite the 30 degree angle. Now, when I do the opposite divided by the hypotenuse, I get 0.5. And this will always be the same for all other right angle triangles with a 30 degree angle. I'm going to add this result to the table I started compiling. So when it's 30 degrees, the opposite side divided by the hypotenuse is always 0.5. I'll also add this result onto the graph I made as well. So back to my theorem with the 60 degree right angle triangle. Seems pretty good, pretty useful to help us solve problems. Unfortunately for me, someone has already made a theorem and far better presented than mine. What's taken me two lines of words can be said with one simple expression, the sine of 60. And that is equal to 0.866025 and so on. This is available on devices each and every one of you own. Type in sine 60 on your calculator and you get 0.866. Not only that, you get a much better and more accurate answer. You will notice I rounded my figures to three decimal places when I measured the triangles earlier, but theirs is far more accurately done than mine. We also looked at triangles which had a 30 degree angle and got 0.5. I also started compiling a table and I got up to 30 and 60 degrees. Again, unfortunately for me, someone has already made this table and has done a far better job with it. I just worked it out with two angles, but they've got values for all possible integer angles from 1 to 89. You might just be able to see the familiar value of 0.866 when it's 60 degrees and 0.5 when it's 30 degrees. And then the graph I started plotting. If you do it for all the angles, you end up with a curve and this starts to form what we call the sine wave. And this is used throughout maths and science. But what about cos and tan? Going back to the very first triangle we measured, we could have done the whole exercise by investigating what happens if we divided the adjacent side by the hypotenuse instead. You'll find you always get 0.5 and writing cos of 60 says the same thing. Doing the exercise with the opposite side divided by the adjacent can be found with tan 60. This is why we teach these formulas in trigonometry to find missing angles or sides. Often using SOH CAH TOA to help students remember them. So in this video, we looked at one very basic way to see where sine, cos and tan come from. Of course, you will know that you can take values of angles like the sine of 270 and the cos of minus 315. But these angles definitely won't work inside of right angle triangles. So in part two, we'll look to make sense of these ideas and introduce a more formal way to define these trig identities using the unit circle. Anyway, thank you for watching. 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