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Trigonometric Formulas


Date: 2/1/96 at 14:27:44
From: Anonymous
Subject: Trigonometric Formulas

My question concerns trigonometric formulas.  How do I show how 
the double angle formulas are derived from the compound angle
formulas?

I think these are the compound angle formulas:

Note: / stands for theta 

1.  cos(/1 + /2) = cos /1 cos /2 - sin /1 sin /2

2.  sin(/1 + /2) = sin /1 cos /2 + cos /1 sin/2

Here are the double angle formulas:

1.  cos 2/ = cos^2 / - sin^2 /

2.  sin 2/ = 2 sin / cos /

I began by trying the compound formula #2

  sin(/1 + /2) =( sin /1 cos /2) +( cos /1 sin/2)

if you multiply the right side you get

=  sin 2/ = sin / cos / + 2sin / + 2 cos / +cos/ sin /


here is where I'm stuck... am I on the right track?

How do I derive the double angle formulas from the compound angle
formulas??

Thanx


Date: 2/1/96 at 16:51:40
From: Doctor Syd
Subject: Re: Trigonometric Formulas

Hello!  

Thanks for writing!  I think you'll be happy to learn that 
deriving the double angle formula from the addition formulas isn't 
as difficult as you may have thought.  Now, I'm not exactly sure 
what you were doing when you took the compound formula #2 and 
"multiplied by the right side."  But, maybe I can give you some 
hints that will lead you in the right direction.  

Let's work through the first addition formula (the one for 
cosine), and then you can try to work through the second on your 
own.  Let t = theta.

Remember that t + t = 2t. 

So, what is the formula for cos (2t)?

Well, cos (2t) = cos (t + t), right?

And, from the addition formula, plug in t for theta 1 and t for 
theta 2 to get:

cos (t + t) = (cos t)(cos t) - (sin t)(sin t)

But, (cos t)(cos t) = (cos t)^2 = cos^2 t

Similarly, (sin t)(sin t) = sin^2 t

So, we get that cos (2t) = cos (t + t) = cos^2 t - sin^2 t 

Thus, we've shown  cos (2t) = cos^2 t - sin^2 t, the double angle 
formula.

See if you can follow similar steps to prove the double angle 
formula for sine!  Good luck, and write back if you have any more 
questions.

-Doctor Syd,  The Math Forum

    
Associated Topics:
High School Trigonometry

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