Difference between revisions of "Trigonometry"

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==Root of Unity==
 
==Root of Unity==
A great way to get into complex numbers (a+b''i'') is via a root of unity, which is a number that satisfies the equation:
+
A great way to get into complex numbers (''a''+''bi'') is via a root of unity, which is a number that satisfies the equation:
  
 
z<sup>n</sup> = 1, where n is a positive integer
 
z<sup>n</sup> = 1, where n is a positive integer
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These roots are:
 
These roots are:
  
e<sup>''2kπi/n''</sup>, where k goes from 0 to n-1 (so there are n roots)
+
e<sup>''2kπi/n''</sup>, where k goes from 1 to n (so there are n roots)
  
 
using Euler's formula gives:
 
using Euler's formula gives:

Latest revision as of 14:24, 1 April 2020

While most people associate trigonometry with triangles, it is more useful in technical fields to consider the trigonometric functions as related to circles.

Sine

F is a function of time F(t) = Asin(ωt + φ)

where:

A, amplitude, the peak deviation of the function from zero.

t, time

ω = 2πf, angular frequency, the rate of change of the function argument in units of radians per second

φ , phase, specifies (in radians) where in its cycle the oscillation is at t = 0.

When φ is non-zero, the entire waveform appears to be shifted in time by the amount φ/ω seconds. A negative value represents a delay, and a positive value represents an advance.


In general, the function may also have a spatial variable x that represents the position on the dimension on which the wave propagates, and a characteristic parameter k called wave number (or angular wave number), which represents the proportionality between the angular frequency ω and the linear speed (speed of propagation) ν; a non-zero center amplitude, D which is

y ( x , t ) = A sin( k x − ω t + φ ) + D, if the wave is moving to the right

y ( x , t ) = A sin( k x + ω t + φ ) + D, if the wave is moving to the left.

The wavenumber is related to the angular frequency by:.

k = ω/v = 2πf/v = 2π/λ

Other Trigonometric Functions

The other trig functions can be defined in terms of the sine function according to the various identity relations.

Euler

A very useful formula for technology is the Euler formula, which relates exponentials, complex numbers and trig functions.

eix = cos x + i sin x

This equation should be memorized since it will be used many times in various forms.


Root of Unity

A great way to get into complex numbers (a+bi) is via a root of unity, which is a number that satisfies the equation:

zn = 1, where n is a positive integer

These roots are:

e2kπi/n, where k goes from 1 to n (so there are n roots)

using Euler's formula gives:

e2kπi/n = cos(2kπ/n) + isin(2kπ/n)

example: roots of x3 are e2iπ/3, e4iπ/3,e6iπ/3

which can be converted to complex numbers:

x = cos(2π/3) + isin(2π/3) = -1/2 + iSqrt(3)/2

x = cos(2π/3) + isin(4π/3) = -1/2 - iSqrt(3)/2

x = cos(2π/3) + isin(6π/3) = 1