When we're confronted with an equation of the form y = sin(x), we can solve it either by using a calculator or recalling the memorized answer. But what can we do when we have an equation of the form x = sin(y)? In this case, the input is a real number, and what we need to find is the angle whose sine equals that real number. For such problems, we use the inverse trigonometric relations.
The inverse trigonometric relations for sine, cosine, tangent, cosecant, secant, and cotangent are, respetively: arcsine, arccosine, arctangent, arccosecant, arcsecant, and arccotangent. Another way to write x = sin(y) is y = arcsin(x). The same holds true for all the inverse relations. Below these six relations are graphed. The graphs of the inverse relations differs from the graphs of the functions only in that the roles of x and y are interchanged.
Note that so far we have referred to these operations as relations. The reason is simple: the operations are not functions. Study the graphs above--do they pass the vertical line test? No. For a given input x, there are either zero, or an infinite number of values of y. This phenomenon is due to the fact that the trigonometric functions are periodic. As an example, let's examine the inverse relation arcsine. What is arcsin(2)? Because there are no angles whose sine is two, no solution exists. How about arcsin()? There are an infinite number of solutions, or angles whose sine is one-half. The domains of the inverse relations are the ranges of their corresponding original functions.
The equation x = sin(y) can also be written y = sin-1(x). This notation can be confusing because though it is meant to express an inverse relationship it also looks like a negative exponent. Nonetheless, it is usually the way that the inverse relations are represented on calculators.
The inverse relations allow us to find values for an unknown angle θ when all we are given is the value of one of the trigonometric functions at the unknown angle. If the ranges of the inverse relations are restricted, they become functions. In the next section, we'll study the inverse trigonometric functions.