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Current time:0:00Total duration:11:07

Review of trig angle addition identities

CCSS.Math:

Video transcript

I've already made a handful of videos that covers what I'm going to cover the trigonometric identities I'm going to cover in this video the reason why I'm doing it is that I'm in need of review myself because I was doing some calculus problems that required me to know this and I have better recording software now so I thought two birds with one stone let me rerecord a video and and kind of refresh things in my own mind so the trig identities that I'm going to assume that we know because I've already made videos on them and they're a little bit involved to remember or to prove are that the sine of a plus B is equal to the sine of a time's the cosine of B plus the sine of B times the cosine of a that's the first one I assume going into this video we know and then if we want to know the sine of well I'll just write it a little differently what if I wanted to figure out the sine of a plus I'll write it this way minus C which is the same thing as a minus C right well we could just use this formula up here to say well that's equal to the sine of a time's the cosine of minus C plus the sine of minus C times the cosine of a and we know and I guess this is another assumption that we're going to have to have going into this video that the cosine cosine of minus C is equal to just the cosine of C that the cosine is an even function and you could look at that by looking at the graph of a of the cosine function or even at the unit circle itself and that the sine is an odd function that the sine of minus C is actually equal to minus sine of C so we can use both of that information to rewrite the second line up here to say that the sine of a minus C is equal to the sine of a time's the cosine of C because the cosine of minus C is the same thing as a cosine of C times the cosine of C and then - the sign of C instead of writing this I can write this - the sign of C times the cosine of a so that we kind of pseudo prove this by knowing this and this ahead of time fair enough and I'm going to use all of these to kind of prove a bunch of more trig identities that I'm going to need so the other eight trig identity is that the cosine the cosine of a plus B is equal to the cosine of a you don't mix up the cosines in the sines in this situation cosine of a times the sine of B and this is - oh sorry I just said you don't mix it up and then I mix them up times the cosine of B minus sine of a times the sine of B now if you want to know what the cosine of a minus B is well you use these same properties cosine of minus B that's just still going to be cosine of B so that's going to be the cosine of a times the cosine cosine a minus B is the same thing as cosine of B and then but here you're going to have sine of minus B which is the same thing as minus sine of B and that minus will cancel that out so it'll be plus sine of a times the sine of B so it's a little tricky when you have a plus sign here you get a minus there when you have a minus sign there you get a plus sign there but fair enough I don't want to dwell on that too much because we have many more identities to show so what if I wanted an identity for let's say the cosine of 2a so the cosine of 2a well that's just the same thing as the cosine of a plus a and then we could use this formula right up here if my second a is just my B then this is just equal to cosine of a time's the cosine of a minus the sine of a times the sine of a write my B is also an a in this case situation which I could rewrite as this is equal to the cosine where'd of a I just wrote cosine of a times itself twice tor times itself - sine squared of a so that's one this is one I guess identity already cosine of 2a is equal to the cosine squared of a minus the sine squared of a let me box off my identities that we're showing in this video so I just showed you that one what if I'm not satisfied what if I just wanted in terms of cosines well we could break out the unit circle definition of our trig functions this is kind of the most fundamental identity that the sine squared of a plus the cosine squared of a is equal to one or you could write that let me think of the best way to do this you could write that the sine squared of a is equal to one minus the cosine squared of a and then we could take this and substitute it right here so we could rewrite this identity as being equal to the cosine squared of a minus the sine squared of a but the sine squared of a is this right there so - I'll do it in a different color - one minus cosine squared of a that's what I just substituted for the sine squared of a and so this is equal to the cosine squared of a minus one plus the cosine squared of a which is equal to let's see we're just adding I'll just continue on the right we have one cosine squared of a plus another cosine squared of a so it's 2 cosine squared of a minus 1 and all of that is equal to cosine of 2a is equal to cosine of 2a now what if I wanted to get an identity that gave me what cosine squared of a is in terms of this so we could just solve for that if we had 1 to both sides of this equation actually let me write this is one of our other identities but if we add one to both sides of that equation we get 2 times the cosine squared of a is equal to cosine of 2a plus 1 and if we divide both sides of this by 2 we get the cosine squared of a is equal to one-half and we could rearrange these I mean just just just just to do it times one plus the cosine of 2a and we're done and we have another identity cosine squared of a that sometimes it's called the power reduction in identity right there now what if we wanted something in terms of the sine squared of a well then maybe we could go back up here and we know from this identity that the sine squared of a is equal to one minus cosine squared of a or we could have gone the other way we could have subtracted sine squared of a from both sides and we could have gotten let me go down there if I subtracted sine squared of Abell from both sides you could get cosine squared of a is equal to one minus sine squared of a sine squared of a and then we could go back into this formula right up here and we could write down and I'll do it in this blue color we could write down to the cosine of 2a the cosine of 2a is equal to instead of writing the cosine squared of a I'll write this is equal to one minus sine squared of a minus sine squared of a sine squared of a and this so this is so my cosine of 2a is equal to C I have a minus sine squared of a minus another sine squared of a so I have 1 minus 2 sine squared of a so here's another identity another way to write my cosine of 2a we're discovering a lot of ways to write our cosine of 2a now if we wanted to solve for sine squared of 2a we could add it to both sides of the equation so let me do that and I'll just write it here for the sake of saving space let me scroll down a little bit and we get so I'm going to go here if I just add sines to sine squared of a to both sides of this I get 2 sine squared of a to sine squared of a plus cosine of two a is equal to one subtract cosine of 2a from both sides you get two sine squared of a is equal to one minus cosine of 2a then you divide both sides of this by 2 and you get sine squared of a is equal to 1/2 times 1 minus cosine of 2a and we have our other our other discovery I guess we could call it our finding and this is interesting it's always interesting to look at the symmetry cosine squared they're identical except for you have a plus 2a here for the cosine squared and you have a minus cosine of 2a here for the sine squared so we've already found a lot of interesting things let's let's see if we can do anything with the sign for the sine of 2a for the sine of 2a so let me pick a new color here that I haven't used well I pretty much used all my colors so if I want to figure out the sine of 2a sine of 2a this is equal to the sine of a plus a which is equal to the sine of a time's the cosine I get that thick times the cosine of a plus and this cosine of a that's the second day X you could view it that way plus the sine I'm just using the sine of a plus B plus the sine of the second day times the cosine of the first day well I just wrote the same thing twice so this is just equal to 2 sine of a cosine of a that was a little bit easier so sine of 2a sine of 2a is equal to that so that's another result so I think I know I'm a little bit tired by playing with all of these sines and cosines and I was able to get all the results that I needed for my calculus problem so hopefully this was a good review for you because it was a good review for me and and you know you can write these things down you can memorize them if you want but the really important takeaway is to realize that you really can derive all of these formulas really from these these initial formulas that we just had and even these I we'll have proofs to show you how to get these from just the basic definitions of your trig functions