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# Intro to the trigonometric ratios

CCSS.Math:

## Video transcript

in this video I want to give you the basics of trigonometry trigonometry and it sounds like a very complicated topic but you're gonna see that it's really just the study of the ratios of sides of triangles the trig part of trigonometry literally means triangle and the metry part literally means measure so let's to me let me just give you some examples here and I think it'll make everything pretty clear so let me draw some right triangles let me just draw one right triangle so this is a right triangle when I say it's a right triangle it's because one of the angles here is 90 degrees this right here is a right that right there is a right angle it is equal to 90 it is equal to 90 degrees and we'll talk about other ways to show the magnitude of angles in future videos so we have a 90 degree angle it's a right triangle let me put some lengths to the sides here so this side over here is maybe three this height right over there is three maybe the base of the triangle right over here is four and then the hypotenuse of the triangle the hypotenuse of the triangle over here is five you only have a hypotenuse when you have a right triangle it is the side opposite the right angle and it is the longest side of a right triangle so that right there is the hypotenuse hypotenuse you probably learned that already from geometry and you can verify that this right triangle decides work out we know from the Pythagorean theorem that 3 squared plus 4 squared plus 4 squared has got to be equal to the length of the longest side the length of the hypotenuse squared is equal to 5 squared so you can verify that these this works out this satisfies the Pythagorean theorem now with that other way let's learn a little bit of trigonometry so the the core core functions of trigonometry we're gonna learn a little bit more about what these functions mean there is the sine the sine function there is the cosine function and there there is the tangent function and you write sine coasts or SI nco and tan for short and these really just specify for any angle in this triangle it'll specify the ratios of certain sides so let me just write something out and this is a little bit of a pneumonic here so something just to help you remember the definitions of these functions but I'm gonna write down something it's called so so so katoa so katoa and you'll be amazed how far this mnemonic will take you in trigonometry so we have sohcahtoa and that what this tells us is so-so tells us that sine sine is equal to opposite over hypotenuse opposite over hypotenuse opposite over hypotenuse it's telling us and this won't make a lot of sense just yet I'll give it a little bit more detail in a second and then cosine is equal to adjacent over hypotenuse cosine is equal to adjacent over hypotenuse and then you finally have tangent you have tangent tangent is equal to opposite over adjacent opposite over adjacent so you're probably saying hey Sal what is this all this opposite hypotenuse adjacent what are we talking about well let's take an angle here let's take an angle here let's say that this angle right over here this angle right over here is Theta between the side of length 4 and the side of length 5 this angle right here is Theta so let's figure out what the sine of theta the cosine of theta and what the tangent of theta are so if we want to first focus on the sine of theta the sine of theta we just have to remember sohcahtoa sohcahtoa sine is opposite over hypotenuse so sine of theta is equal to the opposite so what's the opposite side to the angle so this is our angle right here the opposite side the opposite side if you just go the opposite side so not one of the sides that are kind of adjacent to the angle the opposite side is the three if you're just kind of is opening on to that 3 so the opposite side is 3 and then what's the hypotenuse the hypotenuse well we already know the hypotenuse here is 5 so it's 3 3 over 5 the sine of theta is 3/5 so some since hey what's the sine of that it's 3/5 and we're gonna show I'm going to show you in a second the sine of theta of if this angle is a certain angle it's always going to be 3/5 the ratio of the opposite to the hypotenuse is always going to be the same even if the actual triangle were a larger triangle and or a smaller one so I'll show you that in a second but let's go through all of the trig functions let's think about what the cosine let's think about what the cosine of theta is cosine is adjacent over hypotenuse so remember this let me let me label them we already figured out that the 3 was the opposite side this is the opposite side and only when we're talking about this angle when you talk about this angle this side is opposite to it when you talk about this angle this four side is adjacent to it it's one of the it's one of the sides that kind of make up that that kind of form the vertex here so this right here is an adjacent side adjacent and I want to be very clear this only applies to this angle if we were talking about that angle then this green side would be opposite and this yellow side would be adjacent but we're just focusing on this angle right over here so cosine of this angle we care about adjacent well the adjacent side of this angle is 4 so it is adjacent over the hypotenuse it's the adjacent which is 4 over the hypotenuse 4 over 4 over 5 now let's do the tangent let's do the tangent the tangent of theta opposite over adjacent the opposite side is 3 the opposite side is 3 what is the adjacent side we already figured that out the adjacent side is 4 so knowing the sides of this right triangle we were able to figure out the major trig ratios and we'll see there as there are other trig ratios but they can all be derived from these these three basic trig functions now let's think about another angle in this triangle and I'll redraw it just because my triangle is getting a little bit messy so let's redraw the exact same triangle the exact same triangle and once again the lengths of this triangle are we have a length 4 there we have length 3 there we have length five there the last example we just did we used this data but let's do let's do another angle let's do another angle up here let's do another angle up here and let's call this angle I don't know I'll think of something a random Greek letter so let's say it's zai it's I know it's a little bit of bizarre theta is what you normally use but since I already used theta let's use this I actually even set this to the side let me just simplify it let me call this angle X let's call that angle X so let's figure out the trig functions for that angle X so we have sine of X is going to be equal to what well sine is opposite over hypotenuse so what side is opposite to X well it opens on to this for it opens on to the four so in this context this is now the opposite this is now the opposite side remember 4 was adjacent to this theta but it's opposite to X so it's going to be 4 over now what's the hypotenuse well the hypotenuse is going to be the same regardless of which angle you pick so the hypotenuse is now we're going to be 5 so it's 4/5 now let's do another one what is the cosine what is the cosine of X so cosine is adjacent over hypotenuse what side is adjacent to X that's not the hypotenuse you have the hypotenuse here well the three side it forms it's one of the sides that forms the vertex that X is at and it's not the hypotenuse so this is the adjacent side that is adjacent so it's equal to 3 over the hypotenuse the hypotenuse is 5 and then finally the tangent we want to figure out the tangent of X tangent is opposite over adjacent sohcahtoa tangent is opposite over adjacent opposite over adjacent the opposite side is 4d I want to do it in that blue color the opposite side is 4 and the adjacent side the adjacent side is 3 and we're done in the next video I'll do a ton of more examples of this just so we really get a feel for it but I'll leave you thinking about what happens when these angles start to approach 90 degrees or how could they even get larger 90 degrees and we're going to see is that this definition the sohcahtoa definition takes us a long way for angles that are between zero and 90 degrees or that are less than 90 degrees but they kind of start to mess up really at the boundaries we're going to introduce a new definition that's kind of derived from the sohcahtoa definition for finding the sine cosine and tangent of really any angle