Besides the unit circle method, there is another method for defining (and working with) the basic trigonometric functions.  For this method to work, however, the triangle in question must be a right triangle.  There is a mneumonic (a memory aid) to help you remember the trig functions: SOHCAHTOA (pronounced "so - cuh - toe - uh").  It stands for Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, and Tangent is Opposite over Adjacent.  Remember that Opposite is the length of the side opposite the angle in question, Adjacent is the length of the side adjacent to the angle in question, and Hypotenuse is the length of the hypotenuse of the right triangle.

Here's an example of how to apply SOHCAHTOA.  Refer to the figure shown below.

We use the Pythagorean Theorem to find the length of the missing side.  122 + x2 = 132  simplifies a little to 144 + x2 = 169 and then to x2 = 25.  This says that x = 5 or x = -5.  We choose x = 5 (because of the orientation of the triangle, as will be explained later) and add this information to our drawing:

Looking at the angle labeled θ, the side (leg) opposite θ is 5 long while the hypotenuse is 13 long.  We remember SOHCAHTOA and SOH stands for Sine is Opposite over Hypotenuse.  Therefore, sin(θ) = 5/13.  The side (leg) adjacent to θ is 12 long and the hypotenuse is 13 so, from SOHCAHTOA, cos(θ) = 12/13.  Finally, we get that tangent is opposite over adjacent so tan(θ) = 5/12.  See just how easy SOHCAHTOA is?!

Optional Video Clips

I have found some video clips online (from University of Idaho) that help further explain and give more examples for the trigonometry concepts we've learned in this lesson.  While they are optional, many of them are recommended to help you solidify your understanding of the concepts.  Note that you will need to have a pretty good internet connection for these to work properly.