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#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
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón
#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
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón
#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.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#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.
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 bundy (2013-02-15 01:06:40)
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
#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
#17 2013-02-16 00:10:14
- bob bundy
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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 bundy (2013-02-16 00:14:05)
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei