It has given school kids fits for generations, but trigonometry (the study of the relationships among the sides and interior angles of a triangle) has important applications in the real world. It's also a helpful way to hone one's logical (and geometrical) abilities. Here's a quick overview of the basic principles of trigonometry, and some of its most important results.
Instructions
1. Trigonometry is mostly concerned with "right" triangles. In a right triangle, one of the interior angles is 90 degrees-a property that gives such triangles all sorts of predictable mathematical qualities. Most famously, the Pythagorean Theorem states that the square of the hypotenuse (long side) of a right triangle is equal to the sum of the squares of its other two sides.
2. The three interior angles of a triangle always add up to 180 degrees. This basic property of triangles is key to the rules of trigonometry, and is one of the reasons triangles have such interesting geometric properties. (By the way, this principle only applies to Euclidean triangles; the triangles of non-Euclidean geometries can have interior angles that add up to more or less than 180 degrees.)
3. Trigonometric functions relate the sides and interior angles of a triangle. This is where it gets a bit confusing: When analyzing a right triangle, mathematicians refer to "sine," "cosine" and "tangent" functions (not to mention reciprocals and inverses of these functions, which include "secants" and "arctangents.") The "sine" is the ratio of the side opposite one of the non-90-degree angles to the triangle's hypotenuse; the "cosine" is the ratio of the other side to the hypotenuse; and the "tangent" is the ratio of the triangle's two non-hypotenuse sides.
4. Trigonometric functions have applications that extend beyond right triangles. If you've ever seen a "sine wave"-an undulating, regular, up-and-down wave pattern-you already know that sines, cosines and the rest are often used outside their geometric context. A key property of trigonometric functions is that they're periodic, meaning they repeat after 360 degrees (the circumference of a circle).
5. One of the most important applications of trigonometry is "triangulation." As its name implies, triangulation is the art of determining the distance of objects by calculating the length of the hypotenuse of an imaginary triangle with reference to two fixed points of known distance. This technique is used by architects to survey property and by astronomers to determine the distance of stars.
Tags: interior angles, angles triangle, interior angles triangle, right triangle, important applications, most important, right triangles