Trigonemetric IdentitiesDate: 02/11/2003 at 05:26:40 From: Liza Booth Subject: Trigonemetric identities Hi Dr. Math, Although I have received much help from the FAQ you provide, I am still struggling with the concepts of trigonometric compound angle formulas (e.g. sin(A + B) = sin A cos B + cos A sin B) and products of sums or differences [e.g. 2 sin A cos B = sin(A + B) + sin(A - B)]. Also multiple angles (e.g. sin 2A = 2 sin A cos A) and half angles (e.g. sin A = 2 sin(A/2) cos(A/2) ) have left me confused. Although I can usually look at a question and know where to apply the formulae, I simply don't understand why I am. Is there a connection between these and all the trig identities, etc.? Liza from Sydney, Australia Date: 02/12/2003 at 12:23:37 From: Doctor Peterson Subject: Re: Trigonemetric identities Good question! I generally avoid memorizing any more than I have to, so I look for relations from which I can derive most facts from a few basic ones. The most complicated identities I just have to look up, because I rarely need them. But for the angle sum and difference and double angle identities, I just need to remember two: sin(a + b) = sin(a)cos(b) + cos(a)sin(b) cos(a + b) = cos(a)cos(b) - sin(a)sin(b) (Check those, please - I wrote them from memory!) I remember these by noting that for the sine, I multiply DIFFERENT functions of the two angles, and ADD them; for the cosine I multiply the SAME functions of the two angles, and SUBTRACT them. This is typical: the cosine tends to do everything the opposite of the sine. Now you can get the angle difference identities from these by merely replacing b with -b; knowing that sin(-b) = -sin(b) and cos(-b) = cos(b) helps. The double angle formulas come from replacing b with a. The other identities you quoted, like the one for 2 sin(a)cos(b), come from combining a sum and a difference. To reconstruct this one, just note that we have a product of two DIFFERENT functions, which reminds us of the SINE of the sum and difference. Write out the sum and difference identities for the sine, and note whether you add or subtract them to get what you want: sin(a+b) = sin(a)cos(b) + cos(a)sin(b) sin(a-b) = sin(a)cos(b) - cos(a)sin(b) ------------------- --------------------------- sin(a+b) + sin(a-b) = 2sin(a)cos(b) By adding, we cancel out the cos(a)sin(b) terms. I don't remember the half-angle identities unless I've used them a lot lately. But they can be found by solving the double-angle identities, though it takes more work than the examples above. You'll probably want to memorize them. Of course, for now while you are doing a lot of trig, you may want to have more at your fingertips; I would keep a list of the basic identities with you, and consult that when you need it, reminding yourself to repeat the one you are using a few times to settle it in your mind for next time. When you can't use the list, use the methods I've suggested for reconstructing those you forget. If you have any further questions, feel free to write back. I don't know if I've answered your "why" question, so ask away if I missed the point. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 02/20/2003 at 17:10:13 From: Liza Booth Subject: Thank you (trigonemetric identities) To Dr. Peterson and friends, This is to say, thank you so much for the explanation on trig identities and their relations. You've helped take away the mystery of it all and made it a lot clearer and easier for me to recall. I truly appreciate your excellent response and brilliant service. Your encouragement has helped build my confidence, which I know will make a difference in my final result. It's also very comforting to know that when I get 'stuck' or discouraged, I have Dr. Math to fall back on. Thank you again for your excellence in teaching and I promise that should I get "rich" one day, your company will receive a healthy donation. Kind regards, Liza Booth |
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