Introduction to Inverses of Circular Functions
Each of the circular functions has an inverse: a function that “undoes” whatever the original function does. Defining and working with inverse functions can be tricky.
What Is An Inverse Function?
What is meant by the term inverse function, or the inverse of a function? In general terms, the inverse of a function, if it exists, does exactly the reverse of what the function does. We’ll get more formal in a moment. But first, we must clarify something about notation.
When a function f has an inverse, it is customary to denote it by a superscript, so it reads f –1 . This superscript is not an exponent. The function f –1 is not the same thing as the reciprocal of f . If you see f –1 ( q ) written somewhere, it means the inverse function of f applied to the variable q. It does not mean 1/[f(q)]!
Here is the formal definition. Suppose we have a function f . The inverse of f , call it f –1 , is a function such that f –1 [ f ( x )] = x for all x in the domain of f , and f [ f –1 (y)] = y for all y in the range of f . The function f –1 “undoes” what f does, and the function f “undoes” what f –1 does. If we apply a function to some value of a variable x and then apply the function’s inverse to that, we get x back. If we apply the inverse of a function to some value of a variable y and then apply the original function to that, we get y back.
Not every function has an inverse without some restriction on the domain and/or the range. Sometimes a function f has an inverse f –1 without any restrictions; that is, we can simply turn f “inside-out” and get its inverse without worrying about whether this will work for all the values in the domain and range of f . But often, it is necessary to put restrictions on a function in order to be able to define an inverse. Let’s look at an example.
Square Vs Square Root
Figure 3-8 is a graph of a simple function, f ( x ) = x 2 . In this graph, the values of f ( x ) are plotted on the y axis, so we are graphing the equation y = x 2 . This has a shape familiar to anyone who has taken first-year algebra. It is a parabola opening upward, with its vertex at the origin.
Fig. 3-8. The relation y = x 2 is a function of x.
What do you suppose is the inverse function of f ? You might be tempted to say “The square root.” If you say that, you’re right—partly. Try graphing the parabola with the x and y variables interchanged. You’ll plot the curve for the equation x = y 2 in that case, and you’ll get Fig. 3-9. This is a parabola with exactly the same shape as the one for the equation y = x 2 , but because the x and y axes are switched, the parabola is turned on its side. This is a perfectly good mathematical relation, and it also happens to be a function that maps values of y to values of x. But it is not a function that maps values of x to values of y. If we call this relation, g ( x ) = ± x 1/2 , a function, we are mistaken. We end up with some values of x for which g has no y value (that is okay), and some values of x for which g has two y values (that is not okay). This is easy to see from Fig. 3-9.
Fig. 3-9. The relation y = ± x 1/2 , while the inverse of the function graphed in Fig. 3-8 , is not a function.
What can we do to make g into a legitimate function? We can require that the y values not be negative, and we have a function. Alternatively, we can require that the y values not be positive, and again we have a function. Figure 3-10 shows the graph of y = x 1/2 , with the restriction that y ≥ 0. There exists no abscissa ( x value) that has more than one ordinate ( y value).
Fig. 3-10. The relation y = x 1/2 is a function if we require that y be non-negative.
If you are confused by this, go back to The Cartesian Plane Help and review the distinction between a relation and a function.
Arc What?
We can now define the inverses of the circular functions. There are two ways of denoting an inverse when talking about the sine, cosine, tangent, cosecant, secant, and cotangent. We can use the standard abbreviation and add a superscript –1 after it, or we can write “arc” in front of it. Here are the animals, one by one:
- The inverse of the sine function is the arcsine function. If we are operating on some variable x , the arcsine of x is denoted sin –1 ( x ) or arcsin ( x )
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- The inverse of the cosine function is the arccosine function. If we are operating on some variable x , the arccosine of x is denoted cos –1 ( x ) or arccos ( x )
- The inverse of the tangent function is the arctangent function. If we are operating on some variable x , the arctangent of x is denoted tan –1 ( x ) or arctan ( x )
- The inverse of the cosecant function is the arccosecant function. If we are operating on some variable x , the arccosecant of x is denoted csc –1 ( x ) or arccsc ( x )
- The inverse of the secant function is the arcsecant function. If we are operating on some variable x , the arcsecant of x is denoted sec –1 ( x ) or arcsec ( x )
- The inverse of the cotangent function is the arccotangent function. If we are operating on some variable x , the arccotangent of x is denoted cot –1 ( x ) or arccot ( x )
The sine, cosine, tangent, cosecant, secant, and cotangent require special restrictions in order for the inverses to be definable as legitimate functions. These limits are shown in the graphs of the inverse functions that follow.
Use (and Misuse) Of The –1 Superscript
When using –1 as a superscript in trigonometry, we have to be careful. Ambiguity, or even nonsense, can be the result of improper usage. The expression sin –1 x is not the same thing as (sin x ) –1 . The former expression refers to the inverse sine of x , or the arcsine of x (arcsin x ); but the latter expression means the reciprocal of the sine of x , that is, 1/(sin x ). These are not the same. If you have any question about this, plug in a few numbers and test them.
This brings to light an inconsistency in mathematical usage. It is customary to write (sin x ) 2 as sin 2 x . But don’t try that with the exponent –1, for the reason just demonstrated. You might wonder why the numbers 2 and –1 should be treated so much differently when they are used as superscripts in trigonometry. There is no good answer, except that it is “mathematical convention.”
What about other numbers? Does sin –3 x , for example, mean the reciprocal of the cube of the sine of x , or the cube of the arcsine of x ? Or does it mean the arcsine of the cube of x ? If you are worried that the use of a certain notation or expression might produce confusion, don’t use it. Use something else, even if it looks less elegant. Saying what you mean is more important than conservation of symbols. It is better to look clumsy and be clear and correct, than to look slick and be ambiguous or mistaken.
Inverses of Circular Functions Practice Problems
Practice 1
Is there such a thing as a function that is its own inverse? If so, give one example.
Solution 1
The function f ( x ) = x is its own inverse, and the domain and range both happen to span the entire set of real numbers. If f ( x ) = x , then f –1 (y) = y. To be sure that this is true, we can check to see if the function “undoes its own action,” and that this “undoing operation” works both ways. Let f –1 be the inverse of f . We claim that f –1 [ f ( x )] = x for all real numbers x , and f [ f –1 ( y )] = y for all real numbers y. Checking:
f –1 [ f ( x )] = f –1 ( x ) = x
f [ f –1 ( y )] = f ( y ) = y
It works! In fact, it is almost trivial. Why go through such pains to state the obvious? Well, sometimes the obvious turns out to be false, and the wise mathematician or scientist is always wary of this possibility.
Practice 2
Find another function that is its own inverse.
Solution 2
Consider g ( x ) = 1/ x , with the restriction that the domain and range can attain any real-number value except zero. This function is its own inverse; that is, g –1 ( x ) = 1/ x . To prove this, we must show that g –1 [g( x )] = x for all real numbers x except x = 0, and also that g[g –1 (y)] = y for all real numbers y except y = 0. Checking:
g –1 [g( x )] = g –1 (1/x) = 1/(1/x) = x
g [g –1 ( y )] = g(1/y) = 1/(1/y) = y
It works! This is a little less trivial than the previous example.
Practice 3
Find a function for which there exists no inverse function.
Solution 3
Consider the function h ( x ) = 3 for all real numbers x . If we try to apply this in reverse, we have to set y = 3 in order for h –1 ( y ) to mean anything. Then we end up with all the real numbers at once. Clearly, this is not a function. (Plot a graph of it and see.) Besides this, it is not evident what h –1 ( y ) might be for some value of y other than 3.
Practice Problems for these concepts can be found at: Graphs and Inverses Practice Test
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