A trigonometric equation is any equation that contains a trigonometric function. Up until now we have introduced trigonometric functions, but not fully explore them. In the lessons within this SparkNote on trigonometric equations, we'll learn exactly how to solve trigonometric equations.
As mentioned in Trigonometric Identities, a trigonometric equation that holds true for any angle is called a trigonometric identity. There are other equations, though, that only are true for certain angles. They are generally known as conditional equations, but in this text we'll just call them equations. We'll learn some techniques for solving general equations, as well as how to derive an infinite number of solutions to an equation based on a single solution to that equation.
Only a few simple trigonometric equations can be easily solved without a calculator. Often one might encounter an equation like tan(x) = 3.2. Such an equation has no simple answer that can be memorized. It would be tedious to use a calculator and try numerous values for x until you found one that gave a solution close to 3.2. For problems like these, the inverse trigonometric functions are helpful. The inverse trigonometric functions are the same as the trigonometric functions, except x and y are reversed. For example, another way to say sin(y) = x is y = arcsin(x). The arcsine relation is not a function, though, because it assigns more than one element of the range to each element of the domain. For example, sin(y) = has solutions of y = 30 degrees, 150 degrees, 390 degrees, and so on. When the range is restricted, however, then arcsine is a function, and is written with a capital letter, Arcsine. Using the inverse trigonometric functions, it becomes possible (with a calculator) to solve nearly any trigonometric equation without difficulty.