Difference between revisions of "Trigonometry"
Line 44: | Line 44: | ||
z<sup>n</sup> = 1 | z<sup>n</sup> = 1 | ||
+ | |||
+ | These roots are: | ||
+ | |||
+ | e<sup>2kpi/n</sup> |
Revision as of 13:48, 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 is via a root of unity, which is a number that satisfies the equation:
zn = 1
These roots are:
e2kpi/n