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MATH 1113 PreCalculus
Day #24, Part II
Inverse Trig Functions
If you think about the trig. functions (any of them), you'll recall that one of their most important features is their periodicity. But any function that re-visits the same heights (y-coordinates) over and over again is clearly not one-to-one, so fails the horizontal line test, and does not have an inverse that is a function.
Look at the Geogebra applet for the inverse of the sine function:
The solution is to restrict the domain of the trig function (chopping it off on the left and right), so that it is one to one, and hence, its inverse is a function. Here's a video of the tangent function being restricted to produce an inverse:
Notice, on the video, that the restriction on the DOMAIN (x-axis) of the tangent function produces a restriction on the RANGE (y-axis) of the arctangent function, so that the new (complete) definition of the arctangent function is:
Arctan(x) = y (radians) means y is the angle between -pi/2 and pi/2 whose tangent is x.
Once again, the DOMAIN of the arctangent function is "all real numbers," since the RANGE of the tangent function is "all real numbers." The RANGE of the arctangent function is (artificially) limited to (-pi/2, pi/2) since we restricted the DOMAIN of the tangent function, in order for it to be one-to-one.
A similar argument restricts the sine function in creating the arcsine function:
Arcsin(x) = y (radians) means y is the angle between -pi/2 and pi/2 whose sine is x.
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Notice that this time, the DOMAIN of the arcsine function is [-1, 1], since the RANGE of the sine function is bounded by -1 and 1. On the other hand, the RANGE of the arcsine function is (artificially) limited to [-pi/2, pi/2] since we restricted the DOMAIN of the sine function, in order for it to be one-to-one.
The cosine and secant functions are a bit different, since restricting them to [-pi/2, pi/2] would not produce a one-to-one function, so we have to choose a different restriction. The usual restriction is to the interval [0, pi]. Therefore we have:
Arccos(x) = y (radians) means y is the angle between 0 and pi whose cosine is x.
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Here's a handy table summarizing the domains and ranges of the inverse trig functions:
| Function | Domain | Range |
| Arcsin(x) | [-1, 1] | [-pi/2, pi/2] (radians) [-90o, 90o] (degrees) |
| Arccsc(x) | (-inf, 1] U [1, +inf) | [-pi/2, pi/2] (radians) [-90o, 90o] (degrees) |
| Arctan(x) | All real numbers | (-pi/2, pi/2) (radians) (-90o, 90o) (degrees) |
| Arccot(x) | All real numbers | (0, pi) (radians) * (0, 180o) (degrees) * |
| Arccos(x) | [-1, 1] | [0, pi] (radians) [0, 180o] (degrees) |
| Arcsec(x) | (-inf, 1] U [1, +inf) | [0, pi] (radians) [0, 180o] (degrees) |
*Other restrictions are possible, especially for the Arccotangent, with many authors preferring to have the range of the inverse cotangent to be (-pi/2, 0) U (0, pi/2] (see the MathWorld definitions: http://mathworld.wolfram.com/InverseTrigonometricFunctions.html )
Graphs of the inverse trig. functions: See the graphs at the MathWorld site http://mathworld.wolfram.com/InverseTrigonometricFunctions.html. By far the most important of these is the arctangent function. (I'll soon have a special link up to some important uses for the arctangent function.)
In Calculus, the three inverse trig functions you will use most often are the arcsine, the arctangent and the arcsecant.
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Page Last Modified: 10 March, 2007
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