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# Trig challenge problem: trig values & side ratios

CCSS.Math:

## Video transcript

sort the expressions according to their values you can put any number of cards in a category or leave a category empty and so we have this diagram right over here now we have these cards that have these expressions and we're supposed to we're supposed to sort these into different buckets so we're trying to say well what is the length of segment AC over the set length of segment BC equal to which of these expressions is it equal to and then we should drag it into the appropriate appropriate buckets so to figure these out I've actually already redrawn this problem on my little on my on my little I guess you call it scratch pad or blackboard whatever you want to call it this right over here is that same diagram blown up a little bit here are the expressions that we need to drag into things and here's are the buckets that we need to see which of these expressions are equal to which of these expressions so let's first look at this the length of segment AC over the length of segment BC so let's think about what AC is the length of segment AC AC is this right over here so it's this length this length right over here in purple over the length of segment BC over this length right over here so the ratio is the ratio of the lengths of two sides of a right triangle this is clearly this is clearly a right triangle triangle a B C and I could color that in just so you know what triangle I'm talking about triangle ABC is this entire this entire triangle that we could focus on so you could imagine that it's reasonable that the ratio of two sides of a right triangle are going to be the sine of one of its angles and they give us one of the angles right over here they give us they give us this angle right over here you say well no they just mark that angle but notice one arc is here one arc is here so anywhere we see only one arc that's going to be 30 degrees so this is 30 degrees as well you have two arcs here that's 41 degrees two arcs here this is going to be congruent to that this over here is going to be 41 degrees this is three arcs they don't tell us how many how many degrees that is but this angle with the three arcs is congruent to this angle with the three rocks right over there so anyway this yellow triangle triangle ABC we know this the measure of this angle is 30 degrees and then they give us these two sides so how did these sides late to this 30-degree angle well side AC is adjacent to it it's literally it's one of the sides of the angle that is not the hypotenuse so let me write that down this is adjacent adjacent and what is BC well BC is the hypotenuse of this right triangle it's the side opposite the 90 degrees so this is the hypotenuse so some trig function as when applied to 30 degrees is equal to the adjacent side over the hypotenuse let's write down sohcahtoa just to remind ourselves so so Chi Toa sine of an angle is opposite over hypotenuse cosine of an exit angle is adjacent over hypotenuse so cosine let's write this down cosine of 30 degrees is going to be equal to the length of the edge a of the length of the adjacent side so that is AC AC over the length of the hypotenuse which is equal to BC so this right over here is the same thing this is the same thing as the cosine of 30 degrees so let's drag it in there this is equal to the cosine of 30 degrees now let's look at the next one cosine of angle Dec cosine of angle Dec where is Dec so de c d e C so that's this angle right over here I'll put four arcs here so we don't get it confused so this is angle Dec so what is the cosine of Dec well once again cosine is adjacent over hypotenuse so cosine of angle Dec the adjacent side to this well that's this right over here you might say well isn't this side adjacent well that side side de that is the actual hypotenuse so that's not going to be the adjacent side so the adjacent side is e the adjacent side is I could call it EC it's the length of segment EC and then the hypotenuse the hypotenuse is this right over here is the length of the hypotenuse the hypotenuse is side de or edie however you want to call it and so the length of it is we could just write it as de now what is this also equal to we don't see this choice over here we don't have the ratio EC over de is one of these choices here but what we do have is one of we do get one of the angles here they give us this 41 degrees and the greater of this green side over the length of this green side or this orange side what would that be in terms of if we want to apply a trig function to this angle well relative to this angle the green side is the opposite side and the orange side is still the hypotenuse so relative to 41 degrees so let's write this down it relative to 41 degrees this ratio is the opposite over the hypotenuse it's the cosine of this angle but it's the sine of this angle right over here sine is opposite over hypotenuse so this is equal to the sine of this angle right over here it's equal to the sine of 41 degrees so that is this one right over here the sine of 41 degrees so let's let's drag that into the appropriate bucket so let's let's sine of 41 degrees is the same thing as the cosine of angle Dec I only have two left so now we have to figure out what the sine of angle CDA is so let's see where C da si da is this entire angle it's this entire angle right over here so I could put a bunch of arcs here if I want just to show that it's different than all the other ones so that's that angle right over there so now we're really dealing with this larger right triangle let me highlight that in some let me highlight it in this pink color so we're now dealing with this larger this larger right triangle right over here we care about the sine of this whole thing remember sine is opposite over hypotenuse sine is opposite over hypotenuse so the opposite the opposite side is going to be is going to be C side C a so this is going to be equal to the length of see a over the hypotenuse which is AD over a D so that is going to be that is over a D now once again we don't see that as a choice here but maybe we can express this ratio maybe this ratio is a trig function applied to one of the other angles and they give us one of the angles they give us this angle right over here I guess we could call this angle DAC this is 30 degrees so relative to this angle what two sides are we taking the ratio of we're taking now the ratio of relative to this angle the adjacent side over the hypotenuse so this is the adjacent side over the hypotenuse what deals with adjacent over hypotenuse well cosine so this is equal to the cosine of this angle so this is equal to cosine of 30 degrees sine of CDA is equal to the cosine of this angle right over here so this this one is equal to this right over here so let me let me drag that in so this one is equal to so you can see that I just dragged it in equal to that and now we have one left we have one left homestretch we should be getting excited AE / EB AE let me use this color length of segment AE that's this length right over here let me make that stand out more let me do it in this red this this color right over here that's the length of segment AE over length of segment e B over length of segment EB this is EB right over here this is e B so now we are focused on we're focused on this triangle this right triangle right over here well we know the measure of this angle over here we have double arcs we have double arcs right over right over here and they say this is 41 degrees so we have double marks over here then this is also going to be 41 degrees so relative to this angle what ratio is this is the opposite over over the hypotenuse opposite over the hypotenuse this right over here is going to be sine of that angle sine of 41 degrees so it's equal to this first one right over there so let's let's drag it so this is going to be equal to sine of 41 degrees so none of the ones actually ended up being equal to the tangent of 41 degrees now let's see if we actually got this right I hope I did we did