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Trigonemetric Identities

Date: 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
Associated Topics:
High School Trigonometry

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