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The Sine Addition Formulas - Problem 2
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
We're talking about the sine addition formulas and we have to develop a sine of a difference formula which is going to look something like this sine of theta minus alpha. In order to do this, we need to recall the opposite angle identities for sine an cosine. Recall that the cosine of negative theta equals sine of theta, so opposite inputs give equal outputs for cosine, and for sine opposite inputs give opposite outputs. So sine is an odd function cosine is an even function. Let's see how that works.
Basically I want to be able to express this as the sine of a sum because I already have the sine of a sum formula. So I have sine of beta plus negative alpha that's easy enough and then I just remembered that sine of a sum formula it's sine, cosine, cosine, sine. Sine beta, cosine negative alpha, cosine beta, sine negative alpha. I recall that with the sine formulas the plus sign is preserved, but the sign is the same sine and then use the opposite angle identities to rewrite these guys.
The cosine of negative alpha equals the cosine of alpha but the sine of negative alpha is minus the sine of alpha so I'm going to change signs, cosine theta, sine alpha sine theta and that's your sine of a difference formula sine of beta minus alpha is sine beta cosine alpha minus cosine beta sine alpha.
Now I've got some examples that will allow us to work with formula a little bit. Find the exact values of sine of 15 degrees and sine of -120. Well let's do this one first. You've got to figure out a difference that will give you 15. Well 45 minus 30 will give you 15, oops, sine of 45 minus 30, and sine of the difference sine, cosine, cosine sine. Sine of 45 cosine 30 minus cosine 45 sine 30, I've seen I need a little room I want to move this guy down you don't mind.
Sine of 45 is root 2 over 2 and so is cosine of 45. Cosine of 30 is root 3 over 2, sine of 30 is a half so I have root 6 over 4 minus root 2 over 4 and that's root 6 minus root 2 all over 4 that's my answer, that's the sine of 15 degree.
Now for the sine of -120, there are other ways I could evaluate this I just want to get some practice with the sine of a difference formula. So I'm going to write this as the sine of let's say 60 minus 180 and again you don't absolutely have to evaluate this using the sine of a difference formula, but it's good practice.
So again sine, cosine, cosine, sine; sine 60 cosine 180 minus cosine 60 sine 180. Sine of 60 is root 3 over 2, cosine of 180 remember a quick picture of the unit circle 180 degrees will have us right here with coordinates -1, 0 that's the cosine of 180 and that's the sine of 180. So -1 and 0, so this is -1 and this is 0, so it doesn't really matter what the cosine of 60 is, I don't get to show off that I know it's 1/2 it doesn't matter it's gone and the result is negative root 3 over 2, that's the sine of -120.
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