Math Is Fun Forum

{7/3}

Member

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?

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

Bob

Administrator

Registered: 2010-06-20

Posts: 10,646

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

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Sum and difference of sine and cosine

Hi Bob

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

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“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
The knowledge of some things as a function of age is a delta function.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,646

Re: Sum and difference of sine and cosine

The + sign won't parse for me.

I've edited them to dotplus for now.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

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)

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“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
The knowledge of some things as a function of age is a delta function.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,646

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

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

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.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

{7/3}

Member

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 )amd wrote wrong,I wanted to derive the compound angle formulas using trigonometry.

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

Bob

Administrator

Registered: 2010-06-20

Posts: 10,646

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.

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. 

hint AF = AB - FB

Bob

Last edited by Bob (2013-02-14 02:06:40)

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

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

Bob

Administrator

Registered: 2010-06-20

Posts: 10,646

Re: Sum and difference of sine and cosine

OK.  Let me know how you get on.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

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

Bob

Administrator

Registered: 2010-06-20

Posts: 10,646

Re: Sum and difference of sine and cosine

hi {7/3}

Well done! 

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

{7/3}

Member

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

Bob

Administrator

Registered: 2010-06-20

Posts: 10,646

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

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

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

Bob

Administrator

Registered: 2010-06-20

Posts: 10,646

Re: Sum and difference of sine and cosine

You're welcome. 

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

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob