2.3. TRIGONOMETRIC FUNCTIONS 69
2.3.4 EULER’S IDENTITY
is section contains a computational trick for quickly recovering the various sum-of-angles,
difference-of-angles, and the double angle identities. is section is an enrichment section; it
contains a small amount of something we normally would not cover in a first-year calculus course.
Definition 2.8 i D
p
1.
e problem with the above definition is that it defines something you have been told does not
exist—probably ever since you first encountered square roots. In English, i is the square root of
negative one. e way you deal with this is that i is a number, albeit a funny one, with the added
property that i
2
D 1. For the time being accept it as a notational shortcut. Once we have i
available, one of the great truths of the universe, Euler’s Identity, becomes possible to state.
Knowledge Box 2.25
Euler’s Identity
e
i
D i sin./ C cos./
In order to make use of Euler’s identity we need to know a little bit about complex numbers.
Definition 2.9 A complex number is a number of the form a C bi where a and b are real numbers.
We say that a is the real part of the complex number and b is the imaginary part. If a D 0 the number
is an imaginary number. If b D 0, the number is a plain real number.
Here are the four basic arithmetic operations for complex numbers.
1. .a C bi / C .c C di/ D .a C c/ C .b C d /i
2. .a C bi / .c C di/ D .a c/ C .b d /i
3. .a C bi / .c C di/ D .ac bd/ C .ad C bc/i
4.
.a C bi/
.c C di/
D
ac C bd
c
2
C d
2
C
bc ad
c
2
C d
2
i
One last fact is needed before we can reap the benefits of Euler’s identity. If a C bi D c C di,
then a D c and b D d: In English, if two complex numbers are equal, then their real parts and
their imaginary parts are also equal. On to harvest results!
Example 2.52 In this example we derive the two double angle identities as the real and
imaginary parts of a single expression: