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Using Euler's Formulas to Obtain Trigonometric Identities

Written by tutor Jeffery D.

In this lesson we will explore the derivation of several trigonometric identities, namely

cos (x + y) = cos x cos y - sin x sin y

and

sin (x + y) = sin x cos y + sin x cos y

also

cos 2x = cos2 x - sin2 x

along with

sin 2x = 2 sin x cos x

and lastly, DeMoivre's Formula,

(cos x + i sin x)n = cos nx + i sin nx

using Euler's Formula. To get a good understanding of what is going on, you will need a previous knowledge of series expansions and complex numbers! You may want to refresh your knowledge of those subjects first.

Power series expansions

We start by examining the power series expansion of the functions ex, sin x, and cos x. The power series of a function is commonly derived from the Taylor series of a function for the case where a = 0. This case, where a = 0 is called the MacLaurin Series. The Taylor series:

The case where a = 0 is the MacLaurin Series:

These series are used to approximate the values of functions around a certain point. That is all I'll say about that. The power series of ex; cos x and sin x comes from their MacLaurin Series representation:

for all x.

for all x.

for all x.

Complex numbers and ex

A complex number is a number of the form a + bi where i is a root of the equation x2 + 1 = 0 and a and b are real numbers. Making note of this we can use i in our power series of ex since it is true for all x.

for all x.

Keeping in mind that x2 + 1 = 0 → x = i and so √-1 = ii2 = -1, i3 = -i, etc. So, applying the powers selectively we obtain

We can now rearrange the terms and factor out i so that that we have

Now, if we look back at our series representations of cos x and sin x we have

eix = cos x + i sin x

This conclusion is huge. It is known as Euler's formula. From here we can deduce some of the trigonometric identities as well as come up with formulas for general cases. Let us examine a simple derivation first:

eixeiy = (cos x + i sin x)(cos y + isiny)

But, recall that exey = ex+y. Therefore, we have

eix+iy = cos (x + y) + i sin (x + y) = (cos x + i sin x)(cos y + i sin y)
= cos x cos y + i sin x cos y + i sin y cos x + i2sin x sin y

And now we can rearrange this so that the complex part and the real part is separate.

And so we have

eix+iy = cos (x + y) + i sin (x + y) = (cos x + i sin x)(cos y + i sin y)
= cos x cos y + i sin x cos y + i sin y cos x + i2 sin x sin y
= (cos x cos y - sin x sin y) + (i sin x cos y + i sin y cos x)

Taking the real parts and equating them we obtain the familiar trigonometric sum formula:

cos (x + y) = cos x cos y - sin x sin y

and also

sin (x + y) = sin x cos y + sin y cos x

Now, suppose we have something like this:

eixeix = eix+ix = ei2x = cos (x + x) + i sin(x + x)
= (cos x + i sin x)(cos x + i sin x)
= cos x cos x + i sin x cos x + i sin x cos x + i2 sin x sin x

If we equate the real parts of the equation

cos 2x = cos2x - sin2x

And also we have

sin 2x = 2 sin x cos x

In general, we may obtain a formula for any multiple of an angle in this way. This leads us to another famous formula known as DeMoivre's Formula. DeMoivre's Formula can be derived by taking the nth case of Euler's Formula.

einx = cos nx + i sin nx

We are interested in showing that

(cos x + i sin x)n = cos nx + i sin nx

which is exactly DeMoivre's Formula. It is obvious that this is true for any n. We can show that it is true for all n by using induction.

(cos x + i sin x)n+1 = (cos x + i sin x)n(cos x + i sin x)

From this we apply what we know about the nth case from above.

= (cos nx + i sin nx)(cos x + i sin x)

We can now multiply.

= cos nx cos x + i sin x cos nx + i sin nx cos x + i2 sin nx sin x

And from our work above we have already shown that these can be simplified into our sum formulas as such:

cos (nx + x) + i sin (nx + x) = cos (n + 1)x + i sin (n + 1)x

And so, we have shown that

(cos x + i sin x)n = cos nx + i sin nx

is true for all n. Thus we have shown that some very common trigonometric identities are related and can be derived from series expansions and complex numbers!

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