Trigonometric Functions and Properties

Trigonometric functions are a special set of relationships that are derived from geometric triangles. They are first introduced in high school algebra and geometry, but are used more often in the study of calculus. This article examines the three basic trigonometric functions and highlights their important characteristics.

Sine

The sine function describes the ratio between the opposite and hypotenuse sides of a triangle. The equation for this relationship is written as sin (x) = opposite / hypotenuse. The sine function is a periodic function with a period of 2 * pi (or 360 degrees). The sine function oscillates over its domain and has a range of -1 to 1.

Cosine

The cosine function describes the relationship between the adjacent and hypotenuse sides of a triangle. The formula is written as cos (x) = adjacent / hypotenuse. The cosine function is also a periodic function with a period of 2 * pi (or 360 degrees). The cosine function oscillates just like the sine function, except that it is shifted to the left by 90 degrees. Otherwise the two graphs look identical in shape.

Tangent

The tangent equation describes the ratio between the opposite and adjacent sides of a triangle. The equation for this relationship is written as tan (x) = opposite / adjacent. Tangent is a periodic function with a period of pi (or 180 degrees). The tangent function is in fact the division of the two other trigonometric functions and can be written as tan (x) = sin (x) / cos (x). Tangent does not look like the other graphs since it does not oscillate in a continuous way. Tan (x) is discontinuous at -pi / 2 and pi / 2. At these point the value of tangent is infinite since the value of cos (x) in the quotient is 0. The tangent function has range from negative infinite to positive infinite.

So there are the three basic trigonometric functions. Find some pre calculus practise problems so you can memorize these properties and be ready for differential calculus studies.