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Hovedindhold

# Using trig angle addition identities: finding side lengths

## Video udskrift

what I want to do in this video is use all of our powers all of our knowledge of trig functions and trig identities in order to figure out given all the information that we have here in order to figure out the length of this yellow line this point this line that or the segment that goes from here to here and encourage you to pause the video and think about it before before I work through it so I'm assuming you've had a go at it and in doing that you might have realized it okay this line that's one of the light that's one of the sides of this right triangle that I have right over here and we've given this we're given this alpha and beta but if we consider the combined angle if we consider the combined angle alpha plus beta then this side right over here we can just take out our traditional trig functions so our sohcahtoa definition of the basic trig functions and we know that sine is opposite over hypotenuse so if we're considering alpha plus beta this angle right over here opposite over hypotenuse that's going to be this length over the hypotenuse which is 1 so sine of alpha plus beta is going to be this length right over here so that seems interesting so let me write that down sine of alpha plus beta sine of alpha plus beta plus beta is essentially what we're looking for sine of alpha plus beta is this length right over here sine of alpha plus beta it's equal to the opposite side that over the hypotenuse well the hypotenuse is just going to be equal to 1 so it's equal to this side so another way of phrasing the exact same problem that we first tried to tackle is how do we figure out the sine of alpha plus beta and if you're familiar with you with your trig identities something might be jumping out at you that hey we we know a different way of expressing sine of alpha plus beta we know that this thing is the same thing as we know it's the same thing as the sine of alpha plus our sine of alpha times the cosine of beta plus the other way the cosine of alpha cosine of alpha plus R times the sine of beta times the sine of beta let me draw a line here so we don't get confused so if we're trying to figure this out and we know that this can be re-expressed this way it all boils down to can we figure out what sine of alpha is cosine of beta cosine of alpha and sine of beta now when you look at this you see that you actually can figure those things out so let's do that sine of alpha I'll write it over here sine of alpha is equal to this is alpha sine is opposite over hypotenuse so it's 0.5 over 1 so this is equal to 0.5 so that is 0.5 cosine of beta cosine of beta this is beta cosine is adjacent over hypotenuse so this is beta the adjacent side is 0.6 over the hypotenuse of one so it's 0.6 0.6 0.6 cosine of alpha cosine of alpha adjacent over hypotenuse its square root of 3 over 2 over 1 so that's just square root of 3 over 2 so this is just square root of 3 over 2 and then finally sine of beta sine of beta opposite over hypotenuse is 0.8 this is 0 0.8 and actually let me write that as I'm going to write that as 4/5 just so that that's the same thing as 0.8 just because I think it's going to make it a little bit easier for me to simplify right over here so what is all of this equal to well this is going to be equal to 0.5 times 0.6 this part right over here is 0.3 0.3 and square root of 3 over 2 times 4/5 well let's just multiply them well 4 divided by 2 is 2 so it's 2 square roots of 3 oh for five so this is equal to or so plus two square roots of three over five so this is essentially our answer I feel a little uncomfortable having it in these two different formats where I have a fraction here now but decimal here so let me just write the whole thing as a as a rational expression so zero point three is obviously the same thing as three tenths three tenths so that's the same thing as three over 10 plus now this if I want to write it over ten this is the same thing as four square roots of three over ten and of course if we add these two we are going to get three plus four square roots of 3 plus 4 square roots of 3 all of that over all of that over ten and we are done