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# Graphs of Circular Inverses Help

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By — McGraw-Hill Professional
Updated on Oct 3, 2011

## Graph Of Arcsin Function

Now that you know what the inverse of a function is, we are ready to look at the graphs of the circular inverses, with the restrictions on the domain and the range necessary to ensure that they are legitimate functions.

Figure 3-11 is a graph of the function y = arcsin x (or y = sin –1 x ) with its domain limited to values of x between, and including, –1 and 1 (that is, –1 ≤ x ≤ 1). The range of the arcsine function is limited to values of y between, and including, –90° and 90° (–π/2 rad and π/2 rad).

Fig. 3-11. Graph of the arcsine function for –1 ≤ x ≤ 1.

## Graph Of Arccosine Function

Figure 3-12 is a graph of the function y = arccos x (or y = cos –1 x ) with its domain limited to values of x between, and including, –1 and 1 (that is, –1 ≤ x ≤ 1). The range of the arccosine function is limited to values of y between, and including, 0° and 180° (0 rad and π rad).

Fig. 3-12 . Graph of the arccosine function for –1 ≤ x ≤ 1.

## Graph Of Arctangent Function

Figure 3-13 is a graph of the function y = arctan x (or y = tan –1 x ). The domain encompasses the entire set of real numbers. The range of the arctangent function is limited to values of y between, but not including, –90° and 90° (–π/2 and π/2 rad).

Fig. 3-13. Graph of the arctangent function for –3 ≤ x ≤ 3.

## Graph Of Arccosecant Function

Figure 3-14 is a graph of the function y = arccsc x (or y = csc –1 x ) with its domain limited to values of x less than or equal to –1, or greater than or equal to 1 (that is, x ≤ –1 or x ≥ 1). The range of the arccosecant function is limited to values of y between, and including, –90° and 90° (–π/2 rad and π/2 rad), with the exception of 0° (0 rad). Mathematically, if R represents the range, we can denote it like this in set notation for degrees and radians, respectively:

R = { y : –90° ≤ y < 0° or 0° < y ≤ 90°}

R = { y : –π/2 ≤ y < 0 or 0 < y ≤ π/2}

In the latter expression, the “rad” abbreviation is left out. In pure mathematics, the lack of unit specification for angles implies the use of radians by default. If you see angles expressed in mathematical literature and there are no units specified, you should assume that radians are being used, unless the author specifically states otherwise.

Fig. 3-14. Graph of the arccosecant function for x ≤ –1 and x ≥ 1.

## Graph Of Arcsecant Function

Figure 3-15 is a graph of the function y = arcsec x (or y = sec –1 x ) with its domain limited to values of x such that x ≤ –1 or x ≥ 1. The range of the arcsecant function is limited to values of y such that 0° ≤ y < 90° or 90° < y ≤ 180° (0 rad ≤ y < π/2 rad or π/2 rad < y ≤ π rad).

Fig. 3-15. Graph of the arcsecant function for x ≤ –1 and x ≥ 1.

## Graph Of Arccotangent Function

Figure 3-16 is a graph of the function y = arccot x (or y = cot –1 x ). Its domain encompasses the entire set of real numbers. The range of the arccotangent function is limited to values of y between, but not including, 0° and 180° (0 rad and π rad).

Fig. 3-16. Graph of the arccotangent function for –3 ≤ x ≤ 3.

Practice Problems for these concepts can be found at:  Graphs and Inverses Practice Test

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