A function that has an inverse has exactly one output (belonging to the range) for every input (belonging to the domain), and vice versa. To keep inverse trig functions consistent with this definition, you have to designate ranges for them that will take care of all the possible input values and not have any duplication.
The output values of the inverse trig functions are all angles — in either degrees or radians — and they’re the answer to the question, “Which angle gives me this number?” In general, the output angles for the individual inverse functions are paired up as angles in Quadrants I and II or angles in Quadrants I and IV. The quadrants are selected this way for the inverse trig functions because the pairs are adjacent quadrants, allowing for both positive and negative entries. The notation for these inverse functions uses capital letters.
Domain and range of inverse sine function
The domain for Sin–1x, or Arcsin x, is from –1 to 1. In mathematical notation, the domain or input values, the x’s, fit into the expression
because no matter what angle measure you put into the sine function, the output is restricted to these values. The range, or output, for Sin–1x is all angles from –90 to 90 degrees or, in radians,
If the output is the
then you write these expressions as
The outputs are angles in the adjacent Quadrants I and IV, because the sine is positive in the first quadrant and negative in the second quadrant. Those angles cover all the possible input values.
Domain and range of inverse cosine function
The domain for Cos–1x, or Arccos x, is from –1 to 1, just like the inverse sine function. So the x (or input) values
The range for Cos–1x consists of all angles from 0 to 180 degrees or, in radians,
then you write these expressions as
The outputs are angles in the adjacent Quadrants I and II, because the cosine is positive in the first quadrant and negative in the second quadrant. Those angles cover all the possible input values for the function.
Domain and range of inverse tangent function
The domain for Tan–1x, or Arctan x, is all real numbers — numbers from
This is because the output of the tangent function, this function’s inverse, includes all numbers, without any bounds. The range, or output, of Tan–1x is angles between –90 and 90 degrees or, in radians, between
One important note is that the range doesn’t include those beginning and ending angles; the tangent function isn’t defined for –90 or 90 degrees. The range of Tan–1x includes all the angles in the adjacent Quadrants I and IV, except for the two angles with terminal sides on the y-axis.
Domain and range of inverse cotangent function
The domain of Cot–1x, or Arccot x, is the same as that of the inverse tangent function. The domain includes all real numbers. The range, though, is different — it includes all angles between 0 and 180 degrees
So any angle in Quadrants I and II is included in the range, except for those with terminal sides on the x-axis. Those two angles aren’t in the domain of the cotangent function, so they aren’t in the range of the inverse.
Domain and range of inverse secant function
The domain of Sec–1x, or Arcsec x, consists of all the numbers from 1 on up plus all the numbers from –1 on down. Letting x be the input, you write this expression as
In other words, the domain includes all the numbers from
except for the numbers between –1 and 1. The range of Sec–1x is all the angles between 0 and 180 degrees except for 90 degrees
— meaning all angles in Quadrants I and II, with the exception of 90 degrees, or
Domain and range of inverse cosecant function
The domain of Csc–1x, or Arccsc x, is the same as that for the inverse secant function, all the numbers from 1 on up plus all the numbers from –1 on down. The range is different, though — it includes all angles between –90 and 90 degrees except for 0 degrees or, in radians, between
except for 0 radians.
In short, the range is all the angles in the Quadrants I and IV, with the exception of 0 degrees, or 0 radians.
