Power Reducing Formulas

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This collection of problems explores the power reducing formulas.

Previously we have done things with trigonometric polynomials, such as finding solutions to them or establishing equivalence of expressions. However, this is not the only reasonable thing we might have done. This first problem will serve as a brief tutorial in using the power reducing formulas and the product to sum identities to re-express trigonometric polynomials in sine and cosine to sums of sine and cosine with no powers but different frequencies.

1 :

First, let’s extend the formula for cosine given in the lecture to the third power. A similar process will be required for the remaining two problems.

$\cos (x)^3=\cos (x)\cos (x)^2=\answer {\sage {Ans111}}\cos (x)+\answer {\sage {Ans112}}\cos (2x)$

1.1 :

This is a good start, but generally we are going to want to not have a product of functions in our end result, we just want to have addition. So, we use the product to sum rules to get:

$\cos (x)^3=\answer {\sage {Ans121}}\cos (x)+\answer {\sage {Ans122}}\cos (3x)+\answer {\sage {Ans123}}\cos (-x)$

1.1.1 :

We now use the fact that cosine is an even function and collect like terms to get:

$\cos (x)^3=\answer {\sage {Ans131}}\cos (x)+\answer {\sage {Ans132}}\cos (3x)$

It is important to point out that every possible $\cos (x)^n$ (It is actually worse than this since we can make a power reducing formula for every single pair $\cos (x)^n\sin (x)^m$ .) could have some power reducing formula, but it is not really necessary to make each one of them.

2 : Write $\sage {Disp2}$ as a sum of linear sine and cosine functions. $\sage {Disp2} = \answer {\sage {Ans20}} + \answer {\sage {Ans21}}\cos (x) + \answer {\sage {Ans22}}\cos (2x) + \answer {\sage {Ans23}}\cos (3x) + \answer {\sage {Ans24}}\cos (4x)$

3 : Write $\sage {Disp3}$ as a sum of linear sine and cosine functions. $\sage {Disp3} = \answer {\sage {Ans30}} + \answer {\sage {Ans31}}\sin (x) + \answer {\sage {Ans32}}\cos (2x) + \answer {\sage {Ans33}}\sin (3x) + \answer {\sage {Ans34}}\cos (4x)$

4 : Write $\sage {Disp4}$ as a sum of linear sine and cosine functions. $\sage {Disp4} = \answer {\sage {Ans40}} + \answer {\sage {Ans4S2}}\sin (2x) + \answer {\sage {Ans4C2}}\cos (2x) + \answer {\sage {Ans4S4}}\sin (4x) + \answer {\sage {Ans4C4}}\cos (4x)$