K. Nicole Clark                                                                                            MATH 311-200

2/6/04                                                                                                                   Dr. Fulling

1.1.12 – Revised

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(a) We want to prove

and

starting from

From this equation, we can say that

Using trig identities, this may be rewritten as

thus,

Using the same methods

(b) Using

1.

2.

3.

we want to express

as

and vice versa.

Using 1.,      becomes

To convert constants A and B to C and D, we assume that

which means

Now, going from

to

we will use similar methods.

The equation was rewritten by substituting the identities for cos and sin into the original equation.

Now, we will group the e’s with common exponents together, factoring out the constants.  In order to make the original equation look like the new equation, we will rewrite the constants containing C and D, as constants A and B.

These substitutions allow the final equation to be