Trigonometry is the study of triangles, which contain angles, of course. Get to know some special rules for angles and various other important functions, definitions, and translations. Sines and cosines are two trig functions that factor heavily into any study of trigonometry; they have their own formulas and rules that you’ll want to understand if you plan to study trig for very long.

The graphs of the trig functions have many similarities and many differences. The graphs of the sine and cosine look very much alike, as do the tangent and cotangent, and then the secant and cosecant have similarities. But those three groupings do look different from one another. The one characteristic that ties them all together is the fact that they're periodic, meaning they repeat the same curve or pattern over and over again, in either direction along the x-axis.

The angles used most often in trig have trig functions with convenient exact values. Other angles don’t cooperate anywhere near as nicely as these popular ones do. A quick, easy way to memorize the exact trig-function values of the most common angles is to construct a table, starting with the sine function and working with a pattern of fractions and radicals.

A circle is a geometric figure that needs only two parts to identify it and classify it: its center (or middle) and its radius (the distance from the center to any point on the circle). After you've chosen a point to be the center of a circle and know how far that point is from all the points that lie on the circle, you can draw a fairly decent picture.

The laws of sines and cosines give you relationships between the lengths of the sides and the trig functions of the angles. These laws are used when you don’t have a right triangle — they work in any triangle. You determine which law to use based on what information you have. In general, the side a lies opposite angle A, the side b is opposite angle B, and side c is opposite angle C.

When you know the values for two or more sides of a triangle, you can use the law of cosines. In the following case, you know all three sides (which is called SSS, or side-side-side, in trigonometry) but none of the angles. What you see here is how to solve for the measures of the three angles in triangle ABC, which has sides where a is 7, b is 8, and c is 2.

When you have two sides of a triangle and the angle between them, otherwise known as SAS (side-angle-side), you can use the law of cosines to solve for the other three parts. Consider the triangle ABC where a is 15, c is 20, and angle B is 124 degrees. The following figure shows what this triangle looks like. A sample triangle that allows for the law of cosines.

Identities for angles that are twice as large as one of the common angles (double angles) are used frequently in trig. These identities allow you to deal with a larger angle in the terms of a smaller and more-manageable one. A double-angle function is written, for example, as sin 2θ, cos 2α, or tan 2x, where 2θ, 2α, and 2x are the angle measures and the assumption is that you mean sin(2θ), cos(2α), or tan(2x).

You can use this table of values for trig functions when solving problems, sketching graphs, or doing any number of computations involving trig. The values here are all rounded to three decimal places. θ sinθ cosθ tanθ cotθ secθ cscθ 0° .000 1.000 .000 Undefined 1.000 Undefined 1° .017 1.000 .017 57.290 1.000 57.