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Precalculus: Trigonometric Functions


Inverse Trigonometric Functions

Summary Inverse Trigonometric Functions

The inverses of the trigonometric functions (x = sin(y), x = cos(y), etc.) aren't functions, they are relations. The reason they are not functions is that for a given value of x, there are an infinite number of angles at which the trigonometric functions take on the value of x. Thus, the range of the inverses of the trigonometric functions must be restricted to make them functions. Without these restricted ranges, they are known as the inverse trigonometric relations.

The six inverse trigonometric functions are arcsine, arccosine, arctangent, arccosecant, arcsecant, and arccotangent. Sometimes they are capitalized to differentiate them from the inverse trigonometric relations. In this text, they will not be capitalized; the distinction will be made clear another way. x = sin(y) is equivalent to y = arcsin(x). This is the way that all of the inverse trigonometric functions are defined. In the chart below are shown the domains and ranges of the inverse trigonometric functions.

Figure %: The domains and ranges of the inverse trigonometric functions

Below are pictured the graphs of the six inverse trigonometric relations. Study the restricted ranges that turn these relations into functions, and then locate the sections of the graphs of the relations that become the graphs of the functions. Verify that they are indeed functions.

Figure %: Graph of inverse sine.
Figure %: Graph of inverse cosine.
Figure %: Graph of inverse tangent.
Figure %: Graph of inverse cotangent.
Figure %: Graph of inverse secant.
Figure %: Graph of inverse cosecant.

The inverse trigonometric functions are useful in solving wide varieties of trigonometric functions. On calculators, they appear as sin-1, cos-1, tan-1, etc. This symbolism for the inverse of the functions should not be confused with negative exponents.