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.