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# Prove some properties of the complex sine and cosine functions

The following definitions extend the sine and cosine functions to take arguments :

Prove the following formulas, where are complex numbers and .

1. .
2. .
3. .
4. .
5. .
6. .

1. Proof. Using the given definition of the sine of complex numbers we have

2. Proof. Similar to part (a) we compute

3. Proof. We compute

4. Proof. The two computations are as follows,

5. Proof. We have,

6. Proof. We have,

# Prove the orthogonality relations for sine and cosine using complex exponentials

1. Prove the integral formula,

for integers and .

2. Prove the following orthogonality relations of sine and cosine using the relation in part (a), where and are integers with .

1. Proof. First, if then we have

If then we have

2. Proof. These are all direct computations using part (a). Here they are,

(The final line follows by part (a) and since by hypothesis which implies , and .) Next,

The third formula,

For the next one we use the identities for and derived in this exercise (Section 9.10, Exercise #4(b)).

Finally,

# Prove that sums of trig functions can be expressed as sums of complex exponentials

Prove that sums of the form

are equal to sums of the form

Proof. We compute this directly, substituting in the formulas for sine and cosine in terms of the complex exponential,

that we derived in this exercise. So, we have

where

# Prove DeMoivre’s theorem using complex numbers

1. Prove DeMoivre’s theorem,

for all and all .

2. Prove the triple angle formulas for sine and cosine,

by letting in part (a).

1. Proof. Since we have

2. Letting , we first apply DeMoivre’s theorem to get

On the other hand, we can expand the product,

Equating real and imaginary parts from the two expressions we obtain the requested identities:

# Prove formula relating trig functions of real numbers to the complex exponential

1. Prove that for we have the following formulas,

2. Using part (a) prove that

1. Proof. We compute, using the definition of the complex exponential, :

(Where in the second to last line we used that cosine is an even function and sine is odd, i.e., and .)
For the second formula we compute similarly,

2. Proof. We can compute these directly using the expressions we obtained in part (a),

# Find the limit of the given function

Find the value of the following limit.

Applying L’Hopital’s rule we have,

(We used that from this exercise, Section 7.11, Exercise #1). Since this limit exists, the application of L’Hopital’s rule was justified.

# Find the limit as x goes to 0 of (cos (sin x) – cos x) / x4

Evaluate the limit.

We know (page 287 of Apostol) that the expansions for and as are given by

Therefore, we have the following expansion for as ,

So, now we can take the limit,

# Find the limit as x goes to 0 of (ax – asin x) / x3

Evaluate the limit.

First, we want to get expansions for and as . For we write and use the expansion (page 287 of Apostol) of . This gives us

Next, for , again we write and then use the expansion for we have

Now, we need use the expansion for (again, page 287 of Apostol)

and substitute this into our expansion of ,

(Again, this is the really nice part of little -notation. We had lots of terms in powers of greater than 3, but they all get absorbed into , so we don’t actually have to multiply out and figure out what they all were. We only need to figure out the terms for the powers of up to 3. Of course, the 3 could be any number depending on the situation; we chose 3 in this case because we know that’s what we will want in the limit we are trying to evaluate.)

So, now we have expansions for and (in which most of the terms cancel when we subtract) and we can evaluate the limit.

# Evaluate the limit as x goes to 0 of a given function

Evaluate the limit.

We use the expansions for (given in Example 1 on page 288 of Apostol) and (on page 287 of Apostol):

Therefore, we have

# Find the limit as x goes to 0 of (cosh x – cos x) / x2

Evaluate the limit.

We use the definition of in terms of the exponential:

and the expansions (page 287 of Apostol) of and as :

Putting these together we evaluate the limit: