Euler's formula
Euler's formula is a relationship between exponents of imaginary numbers and the trigonometric functions:
, then
Relationship to sin and cos
In Euler's formula, if we replace θ with -θ in Euler's formula we get
If we add the equations,
and
we get
or equivalently,
Similarly, subtracting
from
and dividing by 2i gives us:
Multiplying the top and bottom by -i gives us:
These formulas allow us to define sin and cos for complex inputs. The existence of these formulas allows us to solve 2nd order differential equations like
y" + y' + y = 0
using sines and cosines.
Example
Find sin(3 + 4i) using Euler's formula:
Using the formula
derived above, we plug 3 + 4i in for θ:
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= |  |
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| = |  |
From Euler's formula,
Plugging these into the formula for sin(3 + 4i) yields:
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Proof of Euler's formula
Given the Maclaurin series for ex, cos(x), and sin(x):
| ex | = |  |
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| = |  |
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| sin(x) | = |  |
|
| = |  |
||
| cos(x) | = |  |
|
| = |  |
Notice that if we plug ix into the Mauclaurin series of ex we get
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After repeated multiplication, i cycles through i, -1, -1, 1, and back again to i, so i has a period of 4. As a result, the terms in the above series switch signs after every 2 terms. If we group the even and odd powers together, we get
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Notice that the terms in parentheses above are equivalent to the Maclaurin series for cos(x) and sin(x) respectively, so plugging cos(x) and sin(x) for their respective Maclaurin series in the above equation yields:
eix = cos(x) + i sin(x)