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The Cosine Addition Formulas - Concept
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
The cosine addition formula calculates the cosine of an angle that is either the sum or difference of two other angles. It arises from the law of cosines and the distance formula. By using the cosine addition formula, the cosine of both the sum and difference of two angles can be found with the two angles' sines and cosines.
I want to derive the cosine of a difference formula but let me explain what that means first. Let's take a look at a picture, this is the unit circle and I have two points a and b and the coordinates of point a are cosine alpha, sine alpha, coordinates of point b are cosine beta, sine beta. I'm interested in finding a formula for the cosine of this angle here beta minus alpha this angle is beta this is alpha so this angle between the two is beta minus alpha I want a cosine for that.
Now let's recall the law of cosines, if I have a triangle and I want to find the length of side c, I need these two sides and the angle between them and I'll use this formula a squared minus b squared minus 2ab cosine of and this is gamma the angle between them.
Alright, let's start by using the law of cosines on this picture, so first law of cosines and let's observe I want solve for this length ab squared that will be our c squared so ab squared equals and then my a and b these two lengths are both 1 because this is the unit circle the radius is 1 so any radius is going to have length 1 both of these have length 1 so it'll be 1 squared plus 1 squared minus 2 times 1 times 1 times the cosine of the angle between them this angle so 1 squared plus 1 squared minus 2 times 1 times 1 times times the cosine of beta minus alpha, beta minus alpha again is this angle. Okay now just simplifying a little bit this is ab squared equals 2 minus 2 cosine of beta minus alpha. Okay so that's one formula.
Now the second thing I want to do is use the distance formula. Remember the distance formula is how we find the distance between two points in the plane and I want to find the distance between these two points actually I want the distance squared. But the distance formula would say that length ab is the square root of and you take the difference in the x coordinates and that's cosine beta minus cosine alpha squared plus the difference in the y coordinates that is sine beta minus sine alpha squared. Now I'm actually interested in the square of this right because eventually I'm going to equate what I get with this so I want ab squared equals and then I'm going to have cosine of beta minus cosine alpha squared plus sine beta minus sine alpha squared. Now we got to expand this, okay it's going to be a mess but prepare yourself, I'm expanding this I get cosine squared beta minus 2 cosine beta cosine alpha plus cosine squared alpha right that's the expansion of this term. Now let's do this term, so plus sine squared beta minus twice the product 2 sine beta sine alpha plus sine squared alpha. It looks terrible but something really really nice is about to happen. Check this out, we got cosine squared beta plus sine squared beta very nice that adds up to one by the Pythagorean identity so I put a 1 down here.
We also have cosine squared alpha and sine squared alpha that also adds up to a 1 so it's another 1, so you write this equal 1+1 and then I have the rest of the stuff the minus 2 let me observe that both of the remaining terms have a minus 2 in front of them so I can write minus 2 I can factor that out and I'll be left with cosine beta cosine alpha right factored out of here and out of here I'll have a plus sine beta sine alpha, so this is precisely 2 minus 2 cosine beta cosine alpha plus sine beta sine alpha. That's what ab squared equals from the distance formula. If we equate it to what ab squared equaled from the law of cosines, we get this right we get this thing equals this 2 minus 2 cosine beta minus alpha. Now let's observe that in both sides both sides of this equation we have 2 minus 2 times something we can cancel these 2's and then divide both sides by negative 2 and what we'll get is this cosine beta cosine alpha plus sine beta sine alpha equals this cosine of beta minus alpha that's our formula bring it right up here. The cosine of beta minus alpha equals cosine beta cosine alpha plus sine beta sine alpha. This is the cosine of the difference formula and we'll use it a lot in coming lessons.
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