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we're on problem 66 and the accompanying diagram measure of angle a is 32 degrees and AC is equal to 10 which equation could be used to find X and triangle ABC so we want X so let's write our sohcahtoa down so Toa what have they given us well X is the opposite side right this is equal to the opposite it's opposite of our angle in question and then this 10 side is the adjacent so I haven't looked at the choices yet but it see if we say the tangent of 32 degrees tangent of 32 degrees should be equal to the opposite side right opposite side which is x over the adjacent side which is 10 and then if we multiply both sides of this by 10 we get 10 let me do it in a darker green 10 tangent of 32 degrees is equal to X and let's hope that's one of that's one of the choices yep sure enough see X is equal to 10 tangent of 32 degrees everything in trigonometry really does boil down to sohcahtoa you'll be amazed when you actually learn take trigonometry as a class you'll see that you know they have all these trig identities and all of that but all of those things can be just proved from sohcahtoa and these are just definitions of ratios that have been very useful in the world when studying triangles all right 67 but you can watch we have a whole playlist on trigonometry and we talk about all the ratios and the unit circle and all of that so if you if you're enjoying these problems I encourage you to watch those videos or if you don't understand these problems I encourage you even more to watch those videos all right the diagram shows an eight-foot ladder leaning against the wall the ladder makes a 53 degree angle with the wall which is closest to the distance up the wall the ladder reaches so that's what this is what they care about this distance right here right so let's see how I would let's see if we how we can figure it out if we have write sohcahtoa maybe we don't have to write that anymore so we have this angle and what have they given us they've given us we want to figure out this deals with the hypotenuse which is 8 it deals with this at inside right so what deals with the this is the adjacent side so what trig function deals with the adjacent side and the hypotenuse well I'll write it down so Toa so we have an innate we want to figure out Jason we have the hypotenuse what trig function deals with it well cosine is adjacent over hypotenuse right so the cosine of 53 degrees the cosine of 53 degrees is equal to the adjacent side of this triangle of this right triangle so that's a and this we should you should only be applying these right now with right triangles later on we'll learn in trigonometry how the trig functions are useful for any triangle but anyway going back to this because obviously only a right triangle has a hypotenuse so you know these H's wouldn't be too meaningful if you're not dealing with a right triangle but anyway back to the problem cosine of 53 it equals the adjacent side over the hypotenuse C the hypotenuse is 8 if we multiply or I could write X there X is the adjacent side right if we multiply both sides of this by 8 we get 8 times the cosine of 53 degrees is equal to X and they tell us let's see they give us a little chart they tell us that the cosine of 53 degrees is approximately equal to 0.6 so this is equal to 0.6 so we get 8 times 0.6 is equal to X 8 times 0.6 is what 4.8 right it's 48 with one digit behind the decimal point so X is equal to 4.8 and that is choice B problem 68 triangle JKL is shown below fair enough which equation should be used to find the length of JK so we want to find this is you know this is the this is J okay all right so we have this angle here we have this angle here JK is what it's the opposite side all right we want to figure out the opposite side we have the hypotenuse so what trig function deals with opposite in hapana and the hypotenuse well I could just write so down I don't have to even write the whole sohcahtoa because sine is equal to the opposite side over the hypotenuse so in this case sine of 24 degrees sine of 24 is equal to the opposite side which is side JK over the hypotenuse this is the hypotenuse right here 28 the longest side of the right triangle and the side that's opposite the 90-degree angle and so maybe we're done cuz it looks like that's how they wrote so sine of 24 degrees is equal to JK over 28 and that is sine sine of 24 is the oh that's choice a is equal to the opposite over the hypotenuse choice a problem 69 these are kind of fun aren't they problem 69 what is the approximate height in feet of the tree in the figure below and now you'll see why trigonometry is useful because I'm sure you you often wonder how tall is that tree and you you know it might be hard to climb it and now you can use trigonometry if you just have a way to measure the angle between your line of sight and the and the and between you know you and the top of the tree and you know you can kind of use you've probably seen those um what do you call those surveyors actually they use things like this and you can figure out the heights of things far away if you know how far away you are from it but anyway let's let's let's work on this problem what is the approximate height and feet of the tree and the figure below okay so this is 50 so let's think about it what do they tell us this is 90 this is 50 so the approximate height and feet so if okay so this is the opposite side of this angle right so let's call this H so what deals with the with the opposite side I should I shouldn't use H because we're let's call that oh oh is the height of the tree because it's opposite the 50 degree angle right so what deals with the opposite and the hypotenuse so once again so katoa right we just have to look at the so part sine is equal to opposite over hypotenuse so the sine of 50 degrees all right here the sine of 50 degrees is equal to the opposite side of this triangle opposite the angle and that's the height of the tree so that's I'll say o for height of tree over the hypotenuse the hypotenuse is 100 feet multiply both sides by 100 you get 100 sine of 50 is equal to this opposite side which is equal to the height of the tree sine of 50 is 0.76 so this is 0.76 so 0.76 times 100 is equal to oh well that's 76 is equal to Oh or the height of the tree and that's choice oh and they said 0.766 so let's put another six so that becomes seventy six point six and that's choice B all right problem 70 these go fast if a is equal to three roots of three in the triangle below so this is equal to three root three I think you're learning a little bit about 30-60-90 triangles now what is the value of B what is the value of B okay maybe they assume that you already do know something about 30-60-90 triangles but anyway if a is equal to three root three what is B and I'm debating on what I should assume that you already know but let's think about it a little bit if I were to draw let me see if how I can do this without resorting to just the definition of 30-60-90 triangles so if I were to draw a so if this angle and I'll prove it for you although later you can memorize because I'm gonna keep saying 30-60-90 triangles we know that this right here is sixty degrees right because 30 plus 60 plus 90 is equal to 100 degrees so that's why we cut we keep calling this a 30-60-90 triangle so how do we figure out this side if we just know if we just know if we just know that side and we know this is 30 degrees so what I'm gonna do is I'm gonna try to redraw this same triangle but I'm gonna flip it over and this is actually the proof of figuring out the measures of 30-60-90 triangle so maybe I'll do it in a different color and I have some time so I'll do it let's see so I'll do it like that then have to bring another line down something like that and then I'll have to draw another line like let's should actually be more like that I could get the idea and then bring a line across like that and actually I'm gonna draw another line down straight down like that okay let's see what we can do with this maybe there's an easier way to do it but this is just what my brain is thinking of right now so let's think about it a little bit in this drawing that I've done this triangle is just a mirror image of that one so this right here this is also 60 right this is 60 this is 60 what's this angle gonna be well there are all collectively supplements of each other right if you go all the way from there to there you have 180 degrees so this big angle right here has got to be 60 degrees sorry 60 degrees this angle right here it's a complement of 30 right so this is 60 degrees and then this is 60 degrees right so this is an equilateral triangle all the sides are going to be equal so that side is equal to that side this is equal to this side all right now let me ask you another question what is what is this side right here what is let me just draw the shorter part and once you memorize 30-60-90 triangles you don't have to go through all of this but it's good to be able to reprove it so what's this length right here well if we look right up there that's length a and this was just a mirror image so this is also length a right so this this whole base of the equilateral triangle is 2a well it's an equilateral triangle so all the sides are the same so this is 2a and this is 2a and just like that we were able to figure out the hypotenuse and they want to know what the B is so once we know two sides of a right triangle it's very easy to figure out the third side so we know that a squared plus C squared is equal a squared plus B squared is equal to C squared let me write that down a squared plus B squared is equal to C squared now what's what's a squared a squared that's right a is 3 roots of 3 so a squared would be 9 times let me write it down 3 roots of three squared plus B squared is equal to let me do it another color what's C C we just figured out it's two times a so it's six roots of three six roots of three squared that's what we did all of this stuff for to figure out that this length is twice that length fair enough now let's simplify so if you take three roots three just the second power the same thing is 3 squared times square root of 3 squared so that's 9 times 3 plus B squared is equal to 36 times 3 and so that's 27 plus B squared is equal to 36 times 3 is 108 108 subtract 27 from both sides B squared is equal to let's see 1 is equal to 81 B is equal to 9 and so that is choice a anyway you should you should watch the videos I've done in the trigonometry playlist on 30-60-90 triangles if you want to be able to do this faster but I think that was useful because you've actually seen how you can how you can figure out the size of 30-60-90 triangle without having memorized ahead of time anyway see in the next video