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

function of time = Asin(2πft + φ) = Asin(ωt + φ)

where:

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

t, time

f, ordinary frequency, the number of oscillations (cycles) that occur each second of 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 = 2 π f v = 2 π λ {\displaystyle k={\omega \over v}={2\pi f \over v}={2\pi \over \lambda }}