# Trigonometry

## Trigonometry Using Circles

For hundreds of years, trigonometry was only considered useful for determining sides and angles of a triangle. However, when mathematicians developed more general definitions for sine, cosine and tangent, trigonometry became much more important in mathematics and science alike. The general definitions for the trigonometric functions were developed by considering these values as points on a unit circle.

A unit circle is one which has a radius of one unit which means x2 + y2 = 1. If we consider the circle to represent the rotation of a side of an angle, then the trigonometric functions can be defined by the x and y coordinates of the point of rotation. For example, coordinates of point P(x,y) can be used to define a right triangle with a hypotenuse of length r. The trigonometric functions could then be represented by the following equations.

With the trigonometric functions defined as such, a graph of each can be developed by plotting its value versus the magnitude of the angle it represents.

Since the value for x and y can never be greater than one on a unit circle, the range for the sine and cosine graphs is between 1 and -1. The magnitude of an angle can be any real number, so the domain of the graphs is all real numbers. (Angles which are greater than 360° or 2π radians represent an angle with more than one revolution of rotation). The sine and cosine graphs are periodic because they repeat their values, or have a period, every 360° or 2π radians. They also have an amplitude of one which is defined as half the difference between the maximum (1) and minimum (-1) values.

Graphs of the other trigonometric functions are possible. Of these, the most important is the graph of the tangent function. Like the sine and cosine graphs, the tangent function is periodic, but it has a period of 180° or π radians. Since the tangent is equal to y/x, its range is - ∞ to ∞ and its amplitude is ∞.

The periodicity of trigonometric functions is more important to modern trigonometry than the ratios they represent. Mathematicians and scientists are now able to describe many types of natural phenomena which reoccur periodically with trigonometric functions. For example, the times of sunsets, sunrises, and comets can all be calculated thanks to trigonometric functions. Also, they can be used to describe seasonal temperature changes, the movement of waves in the ocean, and even the quality of a musical sound.

## Resources

### Books

Barnett, Raymond A., Michael Zeigler, Karl Byleen, and Steven Heath. Analytic Trigonometry with Applications. 7th ed. New York: John Wiley & Sons, 1998.

Blitzer, Robert et al. Algebra and Trigonometry. 2nd ed. Englewood Cliffs, NJ: Prentice Hall, 2003.

Larson, Ron. Calculus With Analytic Geometry. Boston: Houghton Mifflin College, 2002.

Stewart, James, et al. Trigonometry Pacific Grove, CA: Brooks/Cole, 2003.

Weisstein, Eric W. The CRC Concise Encyclopedia of Mathematics. New York: CRC Press, 1998.

Perry Romanowski

## KEY TERMS

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—The side of a right triangle which forms one side of the angle in question.

Amplitude

—A characteristic of a periodic graph represented by half the distance between its maximum and minimum.

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, or the x coordinate of a point on a unit circle.

Degree

—A unit of measurement used to describe the amount of revolution of an angle denoted by the symbol °. There are 360° in a complete revolution.

Hypotenuse

—The longest side of a right triangle which is opposite the right angle.

Law of cosines

—A relationship between the cosine of an angle of a triangle and its sides which can be used to determine the dimensions of a triangle.

Law of sines

—A relationship between the sine of an angle of a triangle and its side which can be used to determine the dimensions of a triangle.

Opposite side

—The side of a right triangle which is opposite the angle in question.

Period

—A value at which a periodic function begins to repeat.

Pythagorean theorem

—An idea suggesting that the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse. It is used to find the distance between two points.

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

Right triangle

—A triangle which contains a 90° or right angle

Similar triangles

—Triangles which have congruent angles and proportional sides.

Sine

—A trigonometric function which represents the ratio of the opposite side of a right triangle to its hypotenuse, or the y coordinate of a point on a unit circle.

Tangent

—A trigonometric function which represents the ratio of the opposite side of right triangle to its adjacent side.

Trigonometric functions

—Angular functions which can be described as ratios of the sides of a right triangle to each other.