SOLUTION: Express cos(3a) in terms of cos⁡(a).

Question 1171119: Express cos(3a) in terms of cos⁡(a).
Answer by ikleyn(52756)   (Show Source): You can put this solution on YOUR website!
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Express cos(3a) in terms of cos⁡(a).
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            There are several ways (more than one unique way) to derive this formula.

            I will show you here very elegant way using complex numbers.

Let z be the complex number z = cos(a) + i*sin(a).

This complex number lies on the unit circle.


Raise the number to degree 3. Using de Moivre formula, you have

     = cos(3a) + i*sin(3a).   (1)


From the other side, applying the binomial expansion formula 

     =  =  = 

                             = .   (2)


In formulas (1) and (2), the left sidea are equal.  Hence, their right sides are equal.


Separating real and imaginary terms in formulas (1) and (2), we get the formula for  


     = .    (3)


Substituting here   = ,  you get from (3)


     =  = .


It is the final formula   = .      ANSWER

Solved.

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