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Trigonometry

Having a basic understanding of trigonometry is essential to understanding how to take data from an image and projecting it back into the real world. Trigonometry is the mathematics behind triangles, and mostly it gives us a few operations we can use on angles or side lengths to solve for the exact dimensions of a triangle.

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SOH CAH TOA

The three basic trig functions are \sin, \cos, and \tan. Given the triangle:

We can define these functions as:

\begin{split} \sin \theta &= \frac{O}{H} \\ \cos \theta &= \frac{A}{H} \\ \tan \theta &= \frac{O}{A} \\ \end{split}

There are also inverse functions for these, so that you can determine the angle from the side lengths. These can be written in one of two ways: putting \text{arc} in front, or by putting a ^{-1} right after the function, such as:

\sin^{-1} \frac{O}{H} = \arcsin \frac{O}{H} = \theta

So as not to get \sin^{-1} x confused with (\sin x)^{-1}, there are another 3 functions that represent the reciprocals of the first three:

\begin{split} \csc \theta = \frac{1}{\sin \theta} = \frac{H}{O} \\ \sec \theta = \frac{1}{\cos \theta} = \frac{H}{A} \\ \cot \theta = \frac{1}{\tan \theta} = \frac{A}{O} \\ \end{split}

all of which can also be inverted by the same methods as above.