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CCSS.Math: ,

we've got two right triangles here let's say we also know that they both have an angle whose measure is equal to theta so angle a is congruent to angle D what do we now know about these two triangles well for any right over any triangle if you know two of the angles you're going to know the third angle because the sum of the angles of a triangle add up to 180 degrees so if you have two angles in common that means you're going to have three angles in common and if you have three angles in common you are dealing with similar triangles let me make that a little bit clearer so if this angle is Theta this is 90 they all have to add up to 180 degrees that means that this angle Plus this angle up here have to add up to 90 we've already used up 90 right over here so this angle angle a and angle B need to be complements so this angle right over here needs to be 90 minus theta well we can use the same logic over here we already used up 90 degrees over here so we have a remaining 90 degrees between theta and that angle so this angle is going to be 90 degrees minus theta 90 degrees minus theta you have three congruent three corresponding angles being congruent you are dealing with similar triangles now why is that interesting well we know from geometry that the ratio of corresponding sides of a similar triangles are always going to be the same so let's explore the corresponding sides here well the side that jumps out when you're dealing with the right triangles the most is always the hypotenuse so this right over here is the hypotenuse this hypotenuse is going to correspond to this hypotenuse right over here and we could write that down this is the hypotenuse of this triangle this is the hypotenuse this is the hypotenuse of that triangle now this side right over here side BC what side does that correspond to well if you look at this triangle you can kind of view it as the side that is opposite this angle theta so it's opposite if you go across the triangle you get there so let's go opposite angle D if you go opposite angle a you get two BC opposite angle D you get two EF you get EF so it corresponds to this side right over here and then finally side AC is the one remaining one we could view it as well there's two sides that make up this angle a one of them is a hypotenuse we could call this maybe the adjacent side to it and though D corresponds to a and so this would be the side that corresponds now the whole reason I did that is to leverage that corresponding sides the ratio between corresponding sides of similar triangles is always going to be the same so for example the ratio between BC and the hypotenuse ba so let me write that down BC over B a B a is going to be equal to EF over e d EF e F the length of segment EF over the length of segment e D over the length of segment e D or we could also write that the length of segment AC so AC over the hypotenuse over this triangle type on over a B is equal to DF over de once again this green side over the orange side these are similar triangles that corresponding to each other so this is equal to this is equal to D F over de over de or we could also say we could keep going but I'll just do another one or we could say that the ratio of this side right over this blue side to the green side so of this triangle BC the length of BC over CA over CA is going to be the same as the blue the ratio between these two corresponding sides the blue over the green EF e F / DF / DF and we got all of this from the fact that these are similar triangles so this is true for all similar triangles or this is true for any right triangle that has an angle they it's going then though those two triangles are going to be similar and all of these ratios are going to be the same well maybe we can give names to these ratios relative to the angle theta so from angle theta is point of view from Thetas point of view I'll write theta right over here or we can just remember that what are these what is the ratio of these two sides well from Thetas point of view that blue side is the opposite side its opposites on the opposite side of the right triangle and then the orange side we've already labeled the hypotenuse so from Thetas point of view this is the opposite side over the hypotenuse and I keep stating from Thetas point of view because that wouldn't be the case for this other angle for angle B from angle B's point of view this is the adjacent side over the hypotenuse and we'll think about that relationship later on but let's just all think of it from Thetas point of view right over here so from Thetas point of view what is this well theta is right over here clearly a B and de a B and de are still the hypotenuse is hypotony I don't know how to say that in plural again and what is AC or and what what is AC and what are DF well these are adjacent to it they're one of the sides to sides that make up this angle that is not the hypotenuse so this we can view as the ratio and either of these triangles between the adjacent side so this is relative once again this is opposite angle B but we're only thinking about angle a right here or angle the angle that measures theta or angle D right over your relative to angle a AC is adjacent relative to angle D DF is adjacent so this ratio right over here is the adjacent over the hypotenuse it's going to be the same relative for any right triangle that has an angle theta in it and then finally this over here this is going to be the opposite side once again this was the opposite side over here this ratio for either right triangle is going to be the opposite side over the adjacent side over the adjacent side and I really want to stress the importance we're going to do many many many more examples of this to make the very concrete but for any right triangle that has an angle theta the ratio between its opposite side and its hypotenuse is going to be the same it comes out of similar triangles you just explore that the ratio between the adjacent side to that angle that is theta and the hypotenuse is going to be the same for any any of these triangles as long as it has that angle theta in it and the ratio relative to the angle theta between the opposite side and the adjacent side between the blue side and the green side is always going to be the same these are similar triangles so given that mathematicians decide to give these things names relative to the angle theta this ratio is always going to be the same so they call this the opposite of our hypotenuse they call this the sine of the angle theta so this is the sine let me do this in a new and a new color this is by definition and we're going to extend this definition in the future this is sine of theta this right over here by definition is the cosine of theta and this right over here by definition is the tangent that by definition is the tangent of theta and a mnemonic that will help you remember this and these really are just definitions people realize well by similar triangles for any angle theta this ratio is always going to be the same for for because of similar triangles for any angle theta this ratio is always going to be same this ratio is always going to be same so let's make these definitions and to help us remember it there's the mnemonic sohcahtoa so I'll write like this so so so is sine is opposite over hypotenuse CUH cosine cosine is adjacent over hypotenuse cosine is adjacent over hypotenuse and then finally tangent tangent is opposite over adjacent tangent is opposite opposite over opposite over adjacent so cut Toa and in future videos we'll actually apply these really these definitions these for these trigonometric functions