Polar- & Rectangular Notation
Polar- & Rectangular Notation are used to describe a sinusoid and it’s frequency. Polar notation describes the sinusoid as a magnitude and a phase (in radians). Rectangular on the other hand, describes a sinusoid in a sine part and a cosine part.
The equations are as follows:
$$ Rectangular: A\cos(x) + B\sin(x) $$ $$ Polar: M\cos(x + \theta) $$
The reason a cosine is used for Polar notation is because a sine function can’t represent the DC offset of a signal, because a sine function with a frequency of 0 will result in all 0 being outputted, where a cosine will output the magnitude continuously.
Conversion
These two equations can be used to describe the same sinusoid where you can convert from Rectangular to Polar using the following equations. $$ M = \sqrt{A^2 + B^2} \newline \theta = \arctan(\frac{B}{A}) $$
Of course you can convert back just as easily using the following equations: $$ A = M\cos(\theta) \newline B = M\sin(\theta) $$
Complex Conjugation
Complex Conjugation is a term for a common operation where the phase of a polar notation is inverted (mostly in relation to Fourier Transforms). This is usually indicated by putting an asterisk next to the variable.
For example: if $X[f]$ consists of $Mag X[f]$ and $Phase X[f]$ then $X^[f]$ consists of $Mag X[f]$ and $-Phase X[f]$. In Rectangular Notation this works by not changing the real part, but changing the sign of the imaginary part: if $X[f]$ consists of $Re X[f]$ and $Im X[f]$ then $X^[f]$ is composed of $Re X[f]$ and $-Im X[f]$.