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#1 2013-02-14 20:41:07

{7/3}
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Sum and difference of sine and cosine

How can I derive sum and difference identities of sine and cosine using trigonometry?


There are 10 kinds of people in the world,people who understand binary and people who don't.

#2 2013-02-14 22:08:44

bob bundy
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Re: Sum and difference of sine and cosine

hi {7/3}

Start with the compound angle formulas:



Funny the + sign won't work in Latex??



adding gives:



If you put P = A +B and Q = A-B, you get:



The others are derived in a similar way.

Bob

Last edited by bob bundy (2013-02-14 22:22:38)


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#3 2013-02-14 22:15:28

anonimnystefy
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Re: Sum and difference of sine and cosine

Hi Bob

Why is "plus" written everywhere "+" should go.


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#4 2013-02-14 22:16:47

bob bundy
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Re: Sum and difference of sine and cosine

The + sign won't parse for me.

I've edited them to dotplus for now.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#5 2013-02-14 22:19:12

anonimnystefy
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Re: Sum and difference of sine and cosine

Hmm... weird!


It's working for me, but only on its own!

Last edited by anonimnystefy (2013-02-14 22:21:16)


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#6 2013-02-14 22:23:45

bob bundy
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Re: Sum and difference of sine and cosine

Yes, me too.  All are acsii 43 so it's not that.

I'll post a note to MIF.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#7 2013-02-14 22:30:50

bobbym
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Re: Sum and difference of sine and cosine

Hi;

I already did in the other thread. It looks like it is a tough nut to crack.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#8 2013-02-14 23:29:37

{7/3}
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Re: Sum and difference of sine and cosine

Sorry,I got confused with names of formulas(my mother-tongue isn't english smile )amd wrote wrong,I wanted to derive the compound angle formulas using trigonometry.


There are 10 kinds of people in the world,people who understand binary and people who don't.

#9 2013-02-15 00:44:52

bob bundy
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Re: Sum and difference of sine and cosine

hi {7/3}

I did wonder if that's what you wanted.  I could remember the formulas I gave easily so I took the line of least effort. 

There are many ways of proving the compound angle formulas.  The trig. approach is the first I met whilst still at school, but I haven't used it since.  So I've had to dig deep in my memory.

http://www.bundy.demon.co.uk/images/seventhirds.gif

So, make a triangle ABC with angle A = alpha and angle B = 90

Make a perpendicular line CD so that angle DCA  = 90.

Choose the position of D so that DAC = beta.

Draw DF perpendicular to AB with F on AB, and finally CE perpendicular to DF with E on DF.

Call the point where DF and AC cross point G.

AGF = CGD = 90 minus alpha so GDC = alpha.



note EF = CB



The cosine formula is very similar.  I'll leave it as an exercise.  smile

hint AF = AB - FB

Bob

Last edited by bob bundy (2013-02-15 01:06:40)


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#10 2013-02-15 03:41:28

{7/3}
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Re: Sum and difference of sine and cosine

Thanks for your help,now i'm going to try to derive the cosine formula(thanks again).


There are 10 kinds of people in the world,people who understand binary and people who don't.

#11 2013-02-15 04:00:58

bob bundy
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Re: Sum and difference of sine and cosine

OK.  Let me know how you get on.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#12 2013-02-15 12:03:29

{7/3}
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Re: Sum and difference of sine and cosine

Hi bob. I have derived the cosine's sum formula,and after trying a little i found the difference formulas too:),plus i know how to derive tangent's formula using sine and cosine


There are 10 kinds of people in the world,people who understand binary and people who don't.

#13 2013-02-15 19:03:15

bob bundy
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Re: Sum and difference of sine and cosine

hi {7/3}

Well done!  smile

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#14 2013-02-15 21:33:24

{7/3}
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Re: Sum and difference of sine and cosine

Quick question:are these formulas used to find half angle formulas?


There are 10 kinds of people in the world,people who understand binary and people who don't.

#15 2013-02-15 22:00:26

bob bundy
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Re: Sum and difference of sine and cosine

hi

If you put alpha = beta = A, you get the double angle formulas.

eg.



Now put theta = 2A and you can re-arrange to get



and so on.

Is that what you wanted?

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#16 2013-02-15 23:08:03

{7/3}
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Re: Sum and difference of sine and cosine

Yeah,thanks you helped me a lot in this post.


There are 10 kinds of people in the world,people who understand binary and people who don't.

#17 2013-02-16 00:10:14

bob bundy
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Re: Sum and difference of sine and cosine

You're welcome.  smile

Bob

One final thought:  The compound angle formulas work whatever the angles used.  They aren't limited to acute angles.  The proof above only works for acute angles; in fact, the diagram breaks down if A + B is over 90.  I think that is why I prefer other proofs that are more general.  The rotational matrix proof is my favourite.

Last edited by bob bundy (2013-02-16 00:14:05)


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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