where \(a_1, \ldots, a_n, b_1, \ldots, b_m\) are constants.
Trigonometric Functions:
There are six trigonometric functions: \(\sin(\theta)\text{,}\)\(\cos(\theta)\text{,}\)\(\tan(\theta)\text{,}\)\(\cot(\theta)\text{,}\)\(\sec(\theta)\text{,}\) and \(\csc(\theta).\)
Trigonometric Inverses:
Each of the trigonometric functions has an associated inverse: \(\inverse\sin(\theta)\text{,}\)\(\inverse\cos(\theta)\text{,}\)\(\inverse\tan(\theta)\text{,}\)\(\inverse\cot(\theta)\text{,}\)\(\inverse\sec(\theta)\text{,}\) and \(\inverse\csc(\theta).\)
Algebraically combining this basic set of fourteen functions allows us to build almost all of the functions you have encountered so far in your mathematics education. Loosely speaking what makes these elementary is that we don’t need Calculus to define them.
But Calculus gives us several different ways to define a multitude of new functions. Each new function is a new tool. New tools give us the means to solve new problems, and a new way to approach old problems. This is one reason that the invention of Calculus was such an important advance.
Functions which require Calculus for their definition are (usually) called analytic functions. The precise definition of analytic actually comes directly from the work we did in Section 7.4 on approximating functions with polynomials. But one way or another the definition of analytic is tied to Calculus.
The exponential and logarithmic functions which we take up next occupy a middle ground between the elementary and the analytic functions. They can be defined without Calculus, which is why you’re probably already familiar with them. But in some ways it is better to use Calculus. The exponential functions give us our first, fairly easy–to–understand introduction to the construction of new functions using Calculus so we will start there.
