Inverse Trigonometric Functions

(rev. 9/1/2000, 9/7/2013)

Just as there are 6 trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent), there are 6 inverse trigonometric functions. The inverse trig functions are known by the names sin^–1 x, cos^–1 x, tan^–1 x, csc^–1 x, sec^–1 x, and cot^–1 x. Mathematicians normally pronounce these (and frequently write them) as arcsin x, arccos x, arctan x, arccsc x, arcsec x, and arccot x, respectively.

By sin^–1 x or arcsin x, we mean “the angle whose sine is x.” The sine function has domain and range [–1, 1]. However, if you look at the graph of y = sin x, you will discover that most of that domain is redundant. We can get all of the range simply by looking at the restricted domain .

Pause for a moment to reflect where we are. We have taken a piece of the sine function having domain and range [–1, 1]. Therefore, in the inverse function that we call y = arcsin x, the domain and range will be switched. Namely, the domain for the arcsin function is [–1, 1] and the range is . (We restrict the domain of the sine function in this way so that the arcsin function will be a true function, not just a relation. In precalculus this process is called a domain restriction. We ensure invertibility of the sine function by restricting its domain.)

When we say cos^–1 x or arccos x, we mean “the angle whose cosine is x.” Like the arcsin function, the arccos function has domain [–1, 1]. However, we must take the range to be in order to cover all possible values. In other words, sin^–1 x is capable of returning a negative angle, but cos^–1 x always returns a nonnegative angle in Quadrant I or II.

When we say tan^–1 x or arctan x, we mean “the angle whose tangent is x.” The most sensible domain restriction for the tangent function is the open interval , since that will generate all real numbers as outputs. Therefore, the arctangent function has domain and range .

Summary:

Function	Domain	Range
pure sine function y = sin x		[–1, 1]
restricted sine function y = sin x, for x in Quadrant IV or I only (i.e., no more than plus or minus 90°)		[–1, 1]
arcsin function y = sin^–1 x	[–1, 1]
pure cosine function y = cos x		[–1, 1]
restricted cosine function y = cos x, for x in Quadrants I or II only (i.e., from 0° through 180°)		[–1, 1]
arccos function y = cos^–1 x	[–1, 1]
pure tangent function y = tan x
restricted tangent function y = tan x, for x in Quadrant IV or I only (i.e., less than plus or minus 90° )
arctan function y = tan^–1 x

The other 3 inverse trig functions, namely arccsc, arcsec, and arccot, use domains and ranges chosen to give good results. It is not necessary to memorize these, since they can always be looked up (and, indeed, may differ slightly from one textbook to another).

A good way to work with inverse trig functions is to get into the habit of using the phrase “the angle whose _____ is _____ .” For example, sin^–1 (–0.2583) means the angle whose sine is –0.2583, and csc^–1 4 means the angle whose cosecant is 4.

Simple examples involving inverse trig functions:
sin^–1 0.5 =
csc^–1 2 =

cot^–1 1 =
cos^–1 2 = DNE

A harder example problem: Compute sec^–1 6.395.
The first thing you notice is that your calculator has no sec^–1 key. The closest thing you have is a cos^–1 key. Try rewriting the problem like this:
sec^–1 6.395 = the angle whose secant is 6.395
            = the angle whose reciprocal of cosine is 6.395
            = the angle whose cosine is 1/6.395
            = cos^–1 (1/6.395)

Finally, you are in a position where your calculator is useful. Make sure your MODE is set to radians, and then simply punch in cos^–1(1/6.395) ENTER. The answer is 1.414, correct to 3 decimal places.