K. Nicole Clark MATH 311-200

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

_{}