Trigonometry

Trigonometry is a branch of mathematics concerned with the relationship between angles and their sides and the calculations based on them. First developed during the third century B.C. as a branch of geometry focusing on triangles, trigonometry was used extensively for astronomical measurements. The major trigonometric functions—including sine, cosine, and tangent—were first defined as ratios of sides in a right triangle. Since trigonometric functions are a natural part of any triangle, they can be used to determine the dimensions of any triangle given limited information.

In the eighteenth century, the definitions of trigonometric functions were broadened by being defined as points on a unit circle. This development allowed the construction of graphs of functions related to the angles they represent, which were periodic. Today, using the periodic (regularly repeating) nature of trigonometric functions, mathematicians and scientists have developed mathematical models to predict many natural periodic phenomena.

Trigonometric functions

The principles of trigonometry were originally developed around the relationship among the sides of a right triangle and its angles. The basic idea was that the unknown length of a side or size of an angle could be determined if the length or magnitude of some of the other sides or angles were known. Recall that a triangle is a geometric figure made up of three sides and three angles, the sum of the angles equaling 180 degrees. The three points of a triangle, known as its vertices, are usually denoted by capital letters.

Words to Know

Adjacent side: The side of a right triangle that forms one side of the angle in question.

Angle: A geometric figure created by two lines drawn from the same point.

Cosine: A trigonometric function that relates the ratio of the adjacent side of a right triangle to its hypotenuse.

Geometry: A branch of mathematics originally developed and used to measure common features on Earth, such as lines, circles, angles, triangles, squares, trapezoids, spheres, cones, and cylinders.

Hypotenuse: The longest side of a right triangle that is opposite the right angle.

Opposite side: The side of a right triangle that is opposite the angle in question.

Periodic function: A function that changes regularly over time.

Radian: A unit of angular measurement that relates the radius of a circle to the amount of rotation of the angle. One complete revolution is equal to 2 π radians.

Right triangle: A triangle that contains a 90-degree or right angle.

Sine: A trigonometric function that represents the ratio of the opposite side of a right triangle to its hypotenuse.

Tangent: A trigonometric function that represents the ratio of the opposite side of right triangle to its adjacent side.

Trigonometric function: An angular function that can be described as the ratio of the sides of a right triangle to each other.

Vertices: The point where two lines come together, such as the corners of a triangle.

The longest side of a right triangle, which is directly across the right angle, is known as the hypotenuse. The sides that form the right angle are the legs of the triangle. For either acute angle (less than 90 degrees) 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 of the right triangle is denoted by a lowercase letter.

Three basic functions—the sine (sin), cosine (cos), and tangent (tan)—can be defined for any right triangle. Those functions are defined as follows:

sin θ = length of opposite side ÷ length of hypotenuse, or a / c

cos θ = length of adjacent side ÷ length of hypotenuse, or b / c

tan θ = length of opposite side ÷ length of adjacent side, or a / b

Three other functions—the secant (sec), cosecant (csc), and cotangent (cot)—can be derived from these three basic functions. Each is the inverse of the basic function. Those inverse functions are as follows:

sec θ = 1/sin θ = c/a

csc θ = 1/cos θ = c/b

cot θ = 1/tan θ = b/a

Periodicity of trigonometric functions

One of the most useful characteristics of trigonometric functions is their periodicity. The term periodicity means that the function repeats itself over and over again in a very regular fashion. For example, suppose that you graph the function y = sin θ . In order to solve this equation, one must express the size of the angle θ in radians. A radian is a unit for measuring the size of the angle in which 1 radian equals 180/ π . (The symbol π [pi] is the ratio of the circumference of a circle to its diameter, and it is always the same, 3.141592+, no matter the size of the circle.)

Applications

The use of trigonometry has expanded beyond merely solving problems dealing with right triangles. Some of the most important applications today deal with the periodic nature of trigonometric functions. For example, the times of sunsets, sunrises, and comet appearances can all be calculated by using trigonometric functions. Such functions also can be used to describe seasonal temperature changes, the movement of waves in the ocean, and even the quality of a musical sound.