# Trigonometry

A list of trigonometric formulae and tables.

Trigonometric functions in polar coordinate form- radius $\rho$, azimuth $\phi$. Hypotenuse is $\rho$, adjacent is x and opposite is y.

$\sin \phi$ and $\cos \phi$ get the $y$ and $x$ components of $\rho$.

$\tan \phi$ and $\sec \phi$ get $y$ and $\rho$ from $x$.

$\cot \phi$ and $\csc \phi$ get $x$ and $\rho$ from $y$.

Trigonometric function values as a function of $\phi$. If the point is 0 or $\infty$, the $\pm$ and $\mp$ signs indicate the value of the function to either side of the specified point. Lower values of $\phi$ indicated by the sign on the top.

Transposed version.

Domain and range tables. Note that the range of the inverse trigonometric functions starts at either $-\frac \pi 2$ or 0, depending on the function.

Reciprocal and quotient identities.

Cofunction identities. All take the form $ \operatorname{f} \left( \frac {\pi} {2} - {\phi} \right) = \operatorname{cof} {\phi}$ and vice versa.

Odd and even function identities. $\cos$ and $\sec$ are even. The rest are odd.

Pythagorean identities. These cover all three sets of functions.

Trigonometric identities.

Sum and difference formulas.

Double angle identities.

Half angle identities.

Power reducing identities.

Spherical coordinates- radius $\rho$, inclination $\theta$, azimuth $\phi$.

Polar form of a complex number.

DeMoivre’s Theorem. $z^n$ is a complex number raised to the power of $n$.

The nth root of a number. There are $n$ complex roots for positive values of $n$. $k$ is used to calculate each root in turn.

Orthogonal decomposition.

Triangulation.

Angles from two locations- $\phi$, $\theta$.
Baseline distance between locations- $\rho$.
Shortest distance from baseline to target object- $y$.
Distance along baseline from respective measuring points to point in front of
target object- $x_\phi$, $x_\theta$.
Distance from respective measuring points to target object- $d_\phi$, $d_\theta$.
**NOTE:** $\phi$ and $\theta$ are interior angles.

Tangent line on circle of radius $\rho$ at angle $\phi$, simple rotations and mirrors.

## Links:

- Wikipedia, List of Trigonometric Identities
- Wikipedia, Polar Coordinate System
- Wikipedia, Spherical Coordinate System
- Domain and Range of the Trigonometric Functions
- Trigonometric Identities
- Product and Sum Formulas
- Polar Form of a Complex Number
- WolframAlpha, Plot
- WolframAlpha, Parametric Plot
- WolframAlpha, Polar Plot
- WolframAlpha, Multi Function Plot
- MathJax Preview