Waste less time on Facebook — follow Brilliant.
×

Trig identities of the complex kind

I'm sure most of you have heard of trig identities involving \(\sin(x)\), \(\cos(x)\),\(\tan(x)\), etc...

But what about complex trigonometry? Specifically the trig function \(\text{cis}(x)\).

This function is short hand for \(\boxed{\cos(x) + i\sin(x)}\) for those of you who haven't met it before. It's used mainly to represent and calculate complex numbers.

So what happens if you add two \(\text{cis}(x)\) functions together? What about subtraction, multiplication, division, how does this all affect the \(\text{cis}(x)\) function?

Before we get started, I'm going to list the identities I already know which involve this function. These and all future identities will be in boxes.

\[\large \boxed{\text{cis}(x) = e^{xi}}\]

\[\large \boxed{(\text{cis}(x))^n = \text{cis}(nx)}\]

Let's first expand on that second one. What if we replace \(n\) with a complex number?

\[(\text{cis}(x))^{a + bi}\]

Knowing that \(\boxed{\text{cis}(x) = e^{xi}}\) really helps here.

\[(e^{xi})^{a + bi}\]

\[e^{(a + bi)xi}\]

\[e^{axi - bx}\]

\[\frac{e^{axi}}{e^{bx}}\]

\[\frac{\text{cis}(ax)}{e^{bx}}\]

\[\large \boxed{(\text{cis}(x))^{a + bi} = \frac{\text{cis}(ax)}{e^{bx}}}\]

So now that that's updated we can go on to find new identities.

We'll start with multiplication.

\[\text{cis}(x)\text{cis}(y)\]

\[e^{xi}e^{yi}\]

\[e^{(x + y)i}\]

\[\text{cis}(x + y)\]

\[\large \boxed{\text{cis}(x)\text{cis}(y) = \text{cis}(x + y)}\]

Now division.

\[\frac{\text{cis}(x)}{\text{cis}(y)}\]

\[\frac{e^{xi}}{e^{yi}}\]

\[e^{xi}e^{-yi}\]

\[e^{(x - y)i}\]

\[\text{cis}(x - y)\]

\[\large \boxed{\frac{\text{cis}(x)}{\text{cis}(y)} = \text{cis}(x - y)}\]

Alright so those are done now. Next is addition and subtraction.

\[\text{cis}(x) + \text{cis}(y)\]

\[\text{cis}(x)\left(1 + \frac{\text{cis}(y)}{\text{cis}(x)}\right)\]

\[\text{cis}(x)(1 + \text{cis}(y - x))\]

\[\large \boxed{\text{cis}(x) + \text{cis}(y) = \text{cis}(x)(1 + \text{cis}(y - x))}\]

And finally subtraction.

\[\text{cis}(x) - \text{cis}(y)\]

\[\text{cis}(x)\left(1 - \frac{\text{cis}(y)}{\text{cis}(x)}\right)\]

\[\text{cis}(x)(1 - \text{cis}(y - x))\]

\[\large \boxed{\text{cis}(x) - \text{cis}(y) = \text{cis}(x)(1 - \text{cis}(y - x))}\]

That's all for now, if there's any you think I've missed, please tell.


Edit: I forgot one: \(\text{arccis}(x)\)

\[\text{cis}(x) = e^{xi}\]

\[\ln{\text{cis}(x)} = xi\]

\[\frac{\ln{\text{cis}(x)}}{i} = x\]

\[\frac{i\ln{\text{cis}(x)}}{-1} = x\]

\[-i\ln{\text{cis}(x)} = x\]

\[x = -i\ln{\text{cis}(x)}\]

Replace \(x\) with \(\text{arccis}(x)\) and \(\text{cis}(x)\) with \(x\)

\[\large \boxed{\text{arccis}(x) = -i\ln{x}}\]


Here's the new list of identities:

\[\large \boxed{\text{cis}(x) = e^{xi}}\]

\[\large \boxed{(\text{cis}(x))^{a + bi} = \frac{\text{cis}(ax)}{e^{bx}}}\]

\[\large \boxed{\text{cis}(x)(\text{cis}(y))^{\pm 1} = \text{cis}(x \pm y)}\]

\[\large \boxed{\text{cis}(x) \pm \text{cis}(y) = \text{cis}(x)(1 \pm \text{cis}(y - x))}\]

\[\large \boxed{\text{arccis}(x) = -i\ln{x}}\]

Note by Jack Rawlin
1 year, 8 months ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

Comments

Sort by:

Top Newest

This is a great introduction to complex trigonometric functions. Can you add that to the page?

Calvin Lin Staff - 1 year, 8 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...