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 function 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).