05 - Sine and Cosine - Definition & Meaning - Part 1 - What is Sin(x) & Cos(x) ?

they're saying ey your force is actually equal to 1. this is

what

's in the x and then y directions now, if your triangle is other than 1, which it always is, then you just take those cut factors and multiply by the hypotenuse, the hypotenuse real that Then you know how much is actually in the x direction. You can do the same in the y direction. Take your cut factor and multiply it by the hypotenuse to get the actual quantity that is for this triangle when it has a length greater than. one is fine and that's why when I take the co

sine

and multiply it by the hypotenuse of these other triangles I get how much there is in that direction, which is the projection, and so on, because when I reduce it like that, that's

what

the sides they actually turn out to be so we've done a ton of material so I want to spend a minute or two here going over it all again because I think it really helps to watch it and listen to it a few times.

Let's say I'm pushing a box at some angle a length of a force of 5 newtons I know a 3 4 5 triangle is special and it's a right triangle the sides of a right triangle I label it there the

sine

is defined as opposite The side of this angle is divided by the hypotenuse, while the

cosine

is defined as the adjacent side divided by exactly the same hypotenuse, so in this case I get 3 over 5. In the other case I get 4 over 5 and it is literally the ratio of how much is up compared to the total force and this is the relationship between how much is horizontal compared to the total force a handy way to think about it is the sign of the angle is the projection towards the x direction and the

cosine

is the projection towards the x direction so sine goes with and cosine always goes with of the total force is in the y direction as a fraction of 0.6 of the total force, another way of saying that it is the sine of 0.6 is called shear function or shear factor in the y direction assuming that the length be 1, then if your length really isn't one just take that ratio and multiply by the actual length of your hypotenuse and you'll get how much is in the y direction, and the same thing here is your cut factor in the x direction if I take That number assuming the length is actually equals 1, that's the amount that would be in the x direction, but my triangle is five times bigger than that, so I multiply it by five and then I get the actual amount in the x direction for my triangle, then we take exactly the same. triangle whose angle we now know is 36.87 degrees and we make it larger so that I am not pushing with 5 newtons, I am pushing with 12. and we do exactly the same calculation if I take the cut factor which is this and I multiply by the hypotenuse I get the amount of force in the y direction 7.2 newtons if I take the shear factor and multiply by the actual hypotenuse then I get exactly how much of this force exists in the x direction the cosine goes with the sine x projection e and cosines of projection y x I get these numbers, I run them through the Pythagorean theorem and find out that yes, this makes a right triangle and then I go and calculate the sine and cosine again using the ratios and I find that the sine and cosine I get matches exactly what I got from the calculator earlier and then we close by saying, let's reduce the triangle so that the real hypotenuse is really just a Newton's law.

We do exactly the same. We take the shear factor, this time multiplied by the hypotenuse we take the shear factor in the x direction multiplied by the hypotenuse and we find that if the hypotenuse is 1 then the y direction is 0.6 newtons and the x direction is 0, 8 newtons, so for every unit length of the hypotenuse this is the The cosine amount is the amount of that force in the x direction and uh, this .6 is the amount of force in the y direction for each unit per every unit of force that you have, that's why when you multiply those shear factors by the hypotenuse that gives you the full projection in those directions, so I really encourage you to watch this twice, it's a lot and it's easy to see and say oh yeah, yeah, I get it, but what's going to happen is we're going to present it. a lot of new concepts and calculating different sides of triangles and then you'll get into more advanced classes and you'll do things with vectors and all that and then maybe you'll know in three months you could say oh, I get it, I know why sine is like that I know why the sine goes in the direction and I know why the cosine goes in the or whatever, you want to take the curved path and you want to split it into different directions so we can analyze the different directions separately, so when we know we have an angle here with some kind of number like a force we want to split the directions to see what happens in those different directions the rest of trigonometry and precalculus and a lot of calculus is really based on this fundamental idea, so look at this a few times, do these calculations yourself and then follow me in the next lesson, we're going to get a lot more practice with the fundamental

meaning

of sine and cosine.