Trigonometry

The triangles used in the previous example were right triangles. During the development of trigonometry, the parts of a right triangle were given certain names. The longest side of the triangle, which is directly across from the right angle, is known as the hypotenuse. The sides that form the right angle, denoted by a box in the diagram, are the legs of the triangle. For either acute angle in the triangle, the leg that forms the angle with the hypotenuse is known as the adjacent side. The side across from this angle is known as the opposite side. Typically, the length of each side is denoted by a lower case letter. In the diagram of triangle ABC, the length of the hypotenuse is indicated by c, the adjacent side is represented by b, and the opposite side by a. The angle of interest is usually represented by θ.

The ratios of the sides of a right triangle to each other are dependent on the magnitude of its acute angles. In mathematics, whenever one value depends on some other value, the relationship is known as a function. Therefore, the ratios in a right triangle are trigonometric functions of its acute angles. Since these relationships are of most importance in trigonometry, they are given special names. The ratio or number obtained by dividing the length of the opposite side by the hypotenuse is known as the sine of the angle θ (abbreviated sin θ). The ratio of the adjacent side to the hypotenuse is called the cosine of the angle θ (abbreviated cos θ). Finally, the ratio of the opposite side to the adjacent side is called the tangent of θ or tan θ. In the triangle ABC, the trigonometric functions are represented by the following equations.

These ratios represent the fundamental functions of trigonometry and should be committed to memory. Many mnemonic devices have been developed to help people remember the names of the functions and the ratios they represent. One of the easiest is the phrase "SOH-CAH-TOA." This means: sine is the opposite over the hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent.

In addition to the three fundamental functions, three reciprocal functions are also defined. The inverse of sin θ, or 1/sin θ, is known as the secant of the angle or sec θ. The inverse of the cos θ is the cosecant or csc θ. Finally, the inverse of the tangent is called the cotangent of cot θ. These functions are typically used in special instances.

The values of the trigonometric functions can be found in various ways. They can often be looked up in tables, which have been compiled over the years. They can also be determined by using infinite series formulas. Conveniently, most calculators and computers have the values of trigonometric functions preprogrammed in.