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2. The Inverse Trigonometric Functions.
If f and g are functions such that f(a) = b if and only if
g(b) = a, then f and g are inverse functions
of each other. For
example, the functions f and g with f(x) = x3 and
are inverse functions of one
another since b = a3 if and only if
[Note that the inverse
function of f(x) = x3 is
while the additive inverse or negative is -f(x) = -x3 and
the multiplicative inverse or reciprocal is 1/f(x) = 1/x3.]
A function f that has the same value b for two different numbers a and c in its domain can not have an inverse function g
since f(a) = b = f(c), with f and g inverses of each other, implies a = g(b) = c. Each of the trigonometric functions sine,
cosine, tangent, cotangent, secant, and cosecant repeats its values in intervals of 
and hence does not have an inverse
function.
However the trigonometric functions with suitably restricted domains have inverses. For example, as x increases from
sin(x) increases steadily from -1 to 1; hence the sine function with domain restricted to the interval
does not repeat values and so has an inverse function. We use Sine (abbreviated Sin) to denote the sine
function with domain the closed interval 
and designate its inverse as Arcsine (Arcsin). The domain of the
Arcsine function is [-1, 1] and its range is 
Similarly, the tangent on the open interval 
takes on all real values once and only once and so has an
inverse. We designate the tangent function with domain restricted to the open interval 
as Tangent (Tan) and its
inverse as Arctangent (Arctan).
As x varies from 0 to 
cos(x) decreases steadily from 1 to -1. Therefore we use Cosine (Cos) to designate the restriction
of the cosine function to the domain 
its inverse is written as Arccosine (Arccos).
The essential facts about these three inverse trigonometric functions are:
y = Arcsin(x) means that x = sin(y) and 
y = Arctan(x) means that x = tan(y) and 
y = Arccos(x) means that x = cos(y) and 
Frequently Arcsin(x), Arctan(x), and Arccos(x) are written sin-1x, tan-1x, and cos-1x respectively. One should be prepared for this bad notation in the literature and not allow it to make one confuse an inverse function with a reciprocal. For example, sin-1x is Arcsin(x) but is not (sin(x))-1 = csc(x).
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