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#1 2013-02-13 21:41:07

{7/3}
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Registered: 2013-02-11
Posts: 210

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.

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#2 2013-02-13 23:08:44

bob bundy
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Registered: 2010-06-20
Posts: 6,308

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 (2013-02-13 23:22:38)


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#3 2013-02-13 23:15:28

anonimnystefy
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Registered: 2011-05-23
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Re: Sum and difference of sine and cosine

Hi Bob

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


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#4 2013-02-13 23:16:47

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

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#5 2013-02-13 23:19:12

anonimnystefy
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From: The Foundation
Registered: 2011-05-23
Posts: 15,359

Re: Sum and difference of sine and cosine

Hmm... weird!


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

Last edited by anonimnystefy (2013-02-13 23:21:16)


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#6 2013-02-13 23:23:45

bob bundy
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Registered: 2010-06-20
Posts: 6,308

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

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#7 2013-02-13 23:30:50

bobbym
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Registered: 2009-04-12
Posts: 84,473

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.

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#8 2013-02-14 00:29:37

{7/3}
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Registered: 2013-02-11
Posts: 210

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.


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#9 2013-02-14 01:44:52

bob bundy
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Registered: 2010-06-20
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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.

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 (2013-02-14 02:06:40)


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

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#10 2013-02-14 04:41:28

{7/3}
Member
Registered: 2013-02-11
Posts: 210

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).


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#11 2013-02-14 05:00:58

bob bundy
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Registered: 2010-06-20
Posts: 6,308

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

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#12 2013-02-14 13:03:29

{7/3}
Member
Registered: 2013-02-11
Posts: 210

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


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#13 2013-02-14 20:03:15

bob bundy
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Registered: 2010-06-20
Posts: 6,308

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

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#14 2013-02-14 22:33:24

{7/3}
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Registered: 2013-02-11
Posts: 210

Re: Sum and difference of sine and cosine

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


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#15 2013-02-14 23:00:26

bob bundy
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Registered: 2010-06-20
Posts: 6,308

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

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#16 2013-02-15 00:08:03

{7/3}
Member
Registered: 2013-02-11
Posts: 210

Re: Sum and difference of sine and cosine

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


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#17 2013-02-15 01:10:14

bob bundy
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Registered: 2010-06-20
Posts: 6,308

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 (2013-02-15 01:14:05)


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

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