Trig identities and examples : Khan Academy on a Stick

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!

Angle addition formulas

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.

Angle addition formula proofs

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

Law of cosines
Law of cosines to determine grade
Law of cosines for star distance
Law of sines
Law of sines for missing angle
Proof of the law of cosines
Proof of the law of cosines to solve for a side of a triangle when 2 sides and an angle are known.
Proof: Law of sines
A simple proof of the Law of Sines

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!

Trigonometry identity review/fun
Trig identities part 2 (part 4 if you watch the proofs)
More playing around with trig identities
Trig identities part 3 (part 5 if you watch the proofs)
Continuation of the playing around with trig identities

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!

2003 AIME II problem 11
A little trigonometry to figure out the area of a triangle
2003 AIME II problem 14
Trigonometry and geometry to find the area of an equilateral (but not regular) hexagon
IIT JEE trigonometry problem 1
2010 IIT JEE Paper I #29 Trigonometry problem
IIT JEE trigonometric maximum
2010 IIT JEE Paper 1 Problem 48 Trigonometric Maximum
IIT JEE trigonometric constraints
2010 IIT JEE Paper 1 Problem 47 Trigonometric Constraints
Trigonometric system example
2010 IIT JEE Paper 1 Problem 55 Trigonometric System

Advanced trig examples

This tutorial is a catch-all for a bunch of things that we haven't been able (for lack of time or ability) to categorize into other tutorials :(