When we introduced the trigonometric functions in Chapter 4, we defined them in general but used them only with acute angles.

In this chapter, we show how these functions are used with angles of any size and with angles measured in radians.

By the mid-1700s, the trigonometric functions had been used for many years as ratios, as in Chapter 4. It was also known that they are useful in describing periodic functions (functions for which values repeat at specific intervals) without reference to triangles. In about 1750, this led the Swiss mathematician Leonhard Euler to include, for the first time in a textbook, the trigonometric functions of numbers (not angles). As we will see in this chapter, this is equivalent to using these functions on angles measured in radians.

Euler wrote over 70 volumes in mathematics and applied subjects such as astronomy, mechanics, and music. (Many of these volumes were dictated, as he was blind for the last 17 years of his life.) He is noted as one of the great mathematicians of all time. His interest in applied subjects often led him to study and develop topics in mathematics that were used in those applications.

Today, trigonometric functions of numbers are of importance in many areas of application such as electric circuits, mechanical vibrations, and rotational motion. Although electronics were unknown in the 1700s, the trigonometric functions of numbers developed at that time played an important role in leading us into the electronic age of today.