# Trigonometry

Big, fancy word, right? Don't be fooled. Looking at the prefix, tri-, you could probably assume that trigonometry ("trig" as it's sometimes called) has something to do with triangles. You would be right! Trig is the study of the properties of triangles. Why is it important? It's used in measuring precise distances, particularly in industries like satellite systems and sciences like astronomy. It's not only space, however. Trig is present in architecture and music, too. Now you may wonder...how is knowing the measurement and properties of triangles relevant to music?? THAT is a great question. Maybe you'll learn the answer from us in these tutorials!
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# Trig identities and examples

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All content in “Trig identities and examples”

## Symmetry and periodicity of trig functions

In this tutorial, we will explore the unit circle in more depth so that we can better appreciate how trig functions of an angle might relate to angles that are in some way symmetric within the unit circle. We'll also look at the periodicity of the functions themselves (why they repeat after a certain change in angle).

## Pythagorean identity

In this tutorial, we look at the relationship between the definitions of sine, cosine and tangent (both SOH CAH TOA and unit circle definitions) and the Pythagorean theorem to derive and apply the Pythagorean identity. This is the building block of much of the rest of the trigonometric identities and will be surprisingly useful the rest of your life!

We'll now see that we can express the sin(a+b) and the cos(a+b) in terms of sin a, sin b, cos a, and cos b. This will be handy in a whole set of applications.

Let's see if we can prove the angle addition formulas for sine and cosine!

## Law of cosines and law of sines

The primary tool that we've had to find the length of a side of a triangle given the other two sides has been the Pythagorean theorem, but that only applies to right triangles. In this tutorial, we'll extend this triangle-side-length toolkit with the law of cosines and the law of sines. Using these tool, given information about side lengths and angles, we can figure out things about even non-right triangles that you may have thought weren't even possible!

## Trigonometric identities

If you're starting to sense that there may be more to trig functions than meet the eye, you are sensing right. In this tutorial you'll discover exciting and beautiful and elegant and hilarious relationships between our favorite trig functions (and maybe a few that we don't particularly like). Warning: Many of these videos are the old, rougher Sal with the cheap equipment!