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The Cosine Addition Formulas - Problem 3
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
Okay here is a harder problem that involves the cosine addition formulas. It says if sine of alpha is 12 or 13 and alpha is between pi over 2 and pi, and cosine of beta is -3 over 5 and beta is between pi and 3 pi over 2, a; find cosine of alpha minus beta and I have over there written and b; find cosine of alpha plus beta. We'll do that in a second.
First of all remember that the cosine addition formulas have both cosine alpha and cosine beta, sine alpha and sine beta, so I just don't need these two values. So first let me draw the angle alpha on the unit circle.
Now it says that alpha is between pi over 2, it's going to be somewhere in this quadrant and that sine of alpha is 12 over 13. Now sine is going to give me the second coordinate of the point on the unit circle, and 12 over 13 is pretty close to 1 so I'm going to put it way up here. So this is my angle alpha and it's going to be something comma 12 over 13, I'll call this x.
Now I need to figure out what x is, x is the cosine of alpha and I'll need it in my formula, so let's use the fact that x² plus 12 over 13² equals what? 1, right? That comes from the fact that x² plus y² equals 1 for the unit circle. This is the unit circle and that's it equation so any point on the unit circle has to satisfy that equation.
All right so we have x² plus 144 over 169 equals, it's 169 over 169 that's what 1. So I subtract, I get x² equals 169 minus 144 it's 25 and that means x is plus or minus 5 over 13 and we know that since we're in the second quadrant, the x coordinate has got to be negative, so we know that we should pick -5 over 13, that gives us the cosine of alpha and we'll need that in a second. Let's do the same thing for beta.
It says cosine of beta is -3 over 5 and beta is between pi and 3 pi over 2. Well that means beta is in the third quadrant between pi and 3 pi over 2 and if the cosine is -3 over 5 that's like -.6, so we have a point about here. -3/5, some y value, this is my angle beta.
I need to find the y value and I'll do it the same way I did it here. I'm going to use the fact that x² plus y² equals 1. So x is minus 3/5 squared plus y² equals and when I square this I'm going to get 25 in the denominator, so I'm going to have 25 over 25. This is 9 over 25 plus y² equals 25 over 25 and then I subtract. Y² is 16 over 25 and that means y is plus or minus 4/5. So which is it?
Well I'm in the third quadrant both sine and cosine are going to be negative here, so the y should be -4/5. Now that took so long I don't remember where I was in the problem, I have to a look again. So here I'm trying to find the cosine of alpha minus beta. Remember in the cosine difference in formula I need cosine of alpha and cosine beta, sine of alpha and sine of beta.
So this is going to be cosine, cosine, sine, sine and remember that the sine changes with the cosine addition formula so minus becomes plus. Now the cosine of alpha is this number, the first coordinate. This is alpha. The first coordinate is -5 over 13. The cosine of beta going back here is the first coordinate of this point, this is angle beta and cosine will be -3/5 plus the sine of alpha, we are given that 12 over 13 and then the sine of beta -4/5 the second coordinates and the rest is just arithmetic with fractions, but arithmetic.
So we're going to have a denominator of, actually let's cancel. Let me hold off on that. So we're going to have a denominator of 15 over 65 minus because I have a minus here 48 over 65. 15 minus 48 is -33 over 65. Now it turns out that that actually doesn't reduce, so it was a good idea not to reduce these fractions ahead of time. That's our final answer for cosine of alpha minus beta. And you'll be glad to know we can use the exact same numbers to come up with the cosine of alpha plus beta let's do that over here.
Remember the formula cosine, cosine, sine, sine. The plus becomes a minus and we just fill in numbers; cosine of alpha -5 over 13, cosine of beta -3 over 5, minus sine of alpha, it's 12 over 13 times the sine of beta -4 over 5. 12 over 13, -4 over 5 and again just like before these are actually the same numbers as before the only difference is this is a minus sign, so we have 15 over 65. We have a minus, minus so plus 48 over 65 and so we just have to add 15 and 48, what's that 63? 63 over 65 and that's our answer for cosine of alpha plus beta.
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