Trigonometry

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.

e^ix = 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, where n is a positive integer

These roots are:

e^2kπi/n, where k goes from 0 to n-1 (so there are n roots)

using Euler's formula gives:

e^2kπi/n = cos(2kπ/n) + isin(2kπ/n)

example: roots of x³ are e^2iπ/3, e^4iπ/3,e^6iπ/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