Difference between revisions of "Trigonometry"
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==Root of Unity== | ==Root of Unity== | ||
| − | A great way to get into complex numbers 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 | + | 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: | ||
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example: | example: | ||
| − | roots of x<sup>3</sup> are ''e''<sup>'' | + | roots of x<sup>3</sup> are ''e''<sup>''2iπ/3''</sup>, ''e''<sup>''4iπ/3''</sup>,''e''<sup>''6iπ/3''</sup> |
| + | |||
| + | which can be converted to complex numbers: | ||
| + | |||
| + | x = cos(2π/3) + isin(2π/3) = -1/2 + ''i''Sqrt(3)/2 | ||
| + | |||
| + | x = cos(2π/3) + isin(4π/3) = -1/2 - ''i''Sqrt(3)/2 | ||
| + | |||
| + | x = cos(2π/3) + isin(6π/3) = 1 | ||
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
