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let's just do a ton of more examples just that we make sure that we're getting this trig function thing down well so let's construct ourselves some right triangles let's construct ourselves some right triangles I want to be very clear the way I have defined it so far this will only work in right triangles so if you're trying to find the trig functions of angles that aren't part of right triangles we're going to see that we're have to construct right triangles but let's just focus on the right triangles for now so let's say that I have a triangle where let's say this length down here is 7 and let's say the length of this side up here let's say that that is 4 and let's figure out what the hypotenuse over here is going to be so we know let's call the hypotenuse H we know that H squared is going to be equal to 7 squared plus 4 squared we know that from the Pythagorean theorem that the hypotenuse squared is equal to the square of each of the the sum of the squares of the other two sides H squared is equal to 7 squared plus 4 squared so this is equal to 49 plus 16 49 plus 16 see 49 plus 10 is fifty nine plus six is 65 it is 65 so this H squared let me write H squared it's a different shade of yellow so we have H squared is equal to 65 to do that right forty-nine plus ten is fifty nine plus another six is sixty-five or we could say that H is equal to if we take the square root of both sides square root square root of 65 and we really can't simplify this at all this is 13 this is the same thing as 13 times 5 both of those are not perfect squares and they're both prime so you can't simplify this any more so this is equal to the square root of 65 now let's find the trig let's find the trig functions for this angle up here let's call that angle up there theta so whatever you do it you always want to write down at least for me it works out to write down sohcahtoa so so though I have these vague memories of my trigonometry teacher maybe I read it in some book I don't know you know some about some type of Indian princess named sohcahtoa or whatever but it's a very useful mnemonic so we can apply sohcahtoa let's find let's say we wanted to find the cosine we want to find the cosine of our angle we want to find the cosine of our angle you say sohcahtoa so the CUH CUH tells us what to do with cosine the cop part tells us that cosine is adjacent over hypotenuse cosine is equal to adjacent over hypotenuse so let's overheat look over here to theta what side is adjacent well we know that the hypotenuse we know that the hypotenuse is this side over here so it can't be that side the only other side that's kind of adjacent to it that isn't the hypotenuse is this 4 so the adjacent side over here that side is it's literally right next to the angle it's one of the sides that kind of forms the angle it's 4 over the hypotenuse the hypotenuse we already know is square root of 65 so it's 4 over the square root of 65 and sometimes people will want you to rationalize the denominator which means they don't like to have an irrational number in the denominator like the square root of 65 and if they if you want to rewrite this without a irrational number in the denominator you can multiply the numerator and the denominator by square root of 65 this clearly will not change the number because we're multiplying it by something over itself so we're multiplying the number by 1 that won't change the number but at least it gets rid of the irrational number in the denominator so the numerator becomes 4 times the square root of 65 and the denominator square root of 65 times square root of 65 is just going to be 65 we didn't get rid of the irrational number it's still there but it's now in the numerator now let's do the other trig functions or let's least the other core trig functions we'll learn in the future that there's actually a ton of them but they're all derived from these so let's think about the sine of theta is once again go to sohcahtoa the so tells us what to do it's sine sine is opposite over hypotenuse sine is equal to opposite over hypotenuse sine is opposite over hypotenuse so for this angle what side is opposite when you just go opposite it when it opens into its opposite the seven so the opposite side is the seven this is right here that is the opposite side and then the hypotenuse it's opposite of our hypotenuse the hypotenuse is the square root of 65 square root of 65 and once again if we wanted to rationalize this we could multiply it times the square root of 65 over the square root of 65 and the numerator will get seven square roots of 65 and in the denominator we will get just 65 again now let's do tangent let us do tangent so if I asked you the tangent of the tangent of theta once again go back to Sokka Toa the Toa part tells us what to do with tangent it tells us it tells us that tangent is equal to opposite over adjacent is equal to opposite over opposite over adjacent so for this angle what is opposite we've already figured it out it's seven it opens into the seven it's opposite to seven so it's seven over what side is adjacent well this four is adjacent this four is adjacent so the adjacent side is four so it's seven over four and we're done we figured out all of the trig ratios for theta let's do another one let's do another one I'll make a little bit concrete because right now we've been saying oh what's tangent of X what's tangent of theta let's make it a little bit more concrete let's say let's say and we draw another right triangle there's another right triangle here everything we're dealing with these are going to be right triangles let's say the hypotenuse has length 4 let's say that this side over here has length 2 and let's say that this length over here is going to be two times the square root of three we can verify that this works if you have this side squared so you have let me write it down two times the square root of three squared plus two squared is equal to what this is - this is going to be four times three four times three plus four and this is going to be equal to twelve plus four is equal to sixteen and 16 is indeed four squared so this does equal four squared it does equal four squared it satisfies the Pythagorean theorem and if you remember some of your work from 30-60-90 triangles that you might have learned in geometry you might recognize that this is a 30-60-90 triangle this right here is our right angle I should have drawn it from the get-go to show that this is a right triangle this angle right over here is our 30-degree angle and then this angle up here this angle up here is a 60-degree angle and it's a 30-60-90 because the side opposite the 30 degrees is half the hypotenuse and then the side opposite the 60 degrees is the square root of 3 times the other side that's not the hypotenuse so with that said we're not gonna this isn't supposed to be a room view of 30-60-90 triangles although I just did it let's actually fought find the trig ratios for the different angles so if I were to ask you or if anyone were to ask you what is what is the sine of 30 degrees and remember the 30 degrees is one of the angles in this triangle but it would apply whenever you have a 30-degree angle and you're dealing with the right triangle we'll have broader definitions in the future but if you say sine of 30 degrees hey this angle right over here is 30 degrees so I can use this right triangle and we just have to remember sohcahtoa let me rewrite it so ha Toa sign tells us so tells us what to do it's sine sine is opposite over hypotenuse sine of 30 degrees is the opposite side that is the opposite side which is 2 over the hypotenuse the hypotenuse here is 4 it is 2/4 which is the same thing as 1/2 sine of 30 degrees you'll see is always going to be equal to 1/2 now what is the cosine what is the cosine of 30 degrees once again go back to sohcahtoa the tells us what to do with cosine cosine is adjacent over hypotenuse so before looking at the 30-degree angle it's the adjacent this right over here is adjacent it's right next to it it's not the hypotenuse it's the adjacent over the hypotenuse so it's two square roots of three adjacent over over the hypotenuse over four or if we simplify that we divide the numerator denominator by two it's the square root of three over two finally let's do the tangent the tangent of 30 degrees we go back to Sokka Toa sohcahtoa Toa tells us what to do with tangent it's opposite over adjacent we go to the 30-degree angle because that's what we care about tangent of 30 tangent of 30 opposite is two opposite is two and the adjacent is two square roots of three it's right next to it it's adjacent to it adjacent means next to so two square roots of three so this is equal to the twos cancel out one over the square root of three or we can multiply the numerator and the denominator by the square root of three so we have square root of three over square root of three and so this is going to be equal to the numerator square root of three and then the denominator right over here is just going to be 3 so that's we've rationalized its square root of 3 over 3 fair enough now let's use the same triangle to figure out the trig ratios for the 60 degrees since we've already drawn it so what is what is the sine of 60 degrees I think you're hopefully getting the hang of it now sine is opposite over adjacent so from the sohcahtoa for the 60 degree angle what side is opposite well it opens out into the two square roots of three so the opposite side is two square roots of three and from the 60 degree angle the adjey or sorry it's opposite of our hypotenuse don't want to confuse you so it's opposite over hypotenuse so it's two square roots of 3 over 4 4 is the hypotenuse so it is equal to this simplifies to square root of three over two what is the cosine of 60 degrees cosine of 60 degrees so remember sohcahtoa cosine is adjacent over hypotenuse adjacent is the two side it's right next to the 60-degree angle so it's two over the hypotenuse which is four so this is equal to one half and then finally what is the tangent what is the tangent of 60 degrees well tangent sohcahtoa tangent is opposite over adjacent opposite the 60 degrees is two square roots of three two square roots of three and adjacent to that adjacent to that is two adjacent to sixty degrees is two so it's opposite over adjacent two square roots of three over two which is just equal to the square root of three and i just want to you know look how these are related the sine of thirty degrees is the same thing as the cosine of 60 degrees the cosine of 30 degrees is the same thing as the sine of 60 degrees and then these guys are the inverse of each other and I think if you think of a little bit about this triangle it'll start to make sense why well keep extending this and give you a lot more practice in the next few videos