Inverses of Trigonometric Functions
Date: 08/07/2002 at 22:59:25 From: Jeremy Beasley Subject: Inverses of Trigonometric Functions I have a question pertaining to the law of inverses and the trignometric functions. In algebra I and II, you learn that functions are defined by applying the "vertical line" test, meaning that for every value "x" there is only one corresponding "y" value. Also, we learned about inverses of functions. To help us understand, the teacher applied the "horizontal line" test to help us determine the possibility of a function having an inverse. If you could draw a horizontal line through a function and the line only intersected once, then it has a possible inverse. However, if the horizontal line intersects twice, making it a secant line, then there is no possible inverse. This test (if it is valid enough), provokes me to question why the trig functions cosine and sine have inverses, yet don't pass the horizontal line test because they are oscillating functions. How is it possible that these functions can have inverses of arcsin and arccos respectively? Are they some type of exception to mathematical law? And if so, why?
Date: 08/08/2002 at 09:49:16 From: Doctor Rick Subject: Re: Inverses of Trigonometric Functions Hi, Jeremy. You've asked a good question. Think about the square root for a moment. The function y = x^2 graphs as a parabola; the line y = c, where c > 0, cuts the graph in two places. Thus this function does not have an inverse. But the square root function is effectively the inverse of the square function. How do we manage this? We limit the domain of the square function to non- negative values. In other words, we define a new function: y = x^2, x >= 0 undefined, x < 0 which graphs as just the right half of the parabola. This function has an inverse: the square root function. The graph of the square root function is a parabola turned on its side - but only the top half of this parabola. This is a source of confusion for many, as these items in the Dr. Math archives indicate: Square Root Function http://mathforum.org/library/drmath/view/52645.html Square Root of 100 http://mathforum.org/library/drmath/view/52650.html We do essentially the same thing with trigonometric functions. They are not invertible as they stand, so we define new functions that are only defined on a specified domain: y = sin(x), -pi/2 <= x <= pi/2 ===> x = arcsin(y) y = cos(x), 0 <= x <= pi ===> x = arccos(y) y = tan(x), -pi/2 < x < pi/2 ===> x = arctan(y) y = cot(x), 0 < x < pi ===> x = arccot(y) y = sec(x), 0 <= x < pi/2 ===> x = arcsec(y) or -pi <= x < pi/2 y = csc(x), 0 < x <= pi/2 ===> x = arccsc(y) or -pi < x <= -pi/2 The limited functions are invertible, and their inverses are arcsin, arccos, etc. We may want to refer to ANY angle x whose sine is y, etc. The notation arcsin(y) is sometimes used in this way. When it is, it does not represent a function; the term PRINCIPAL VALUE is used for the single value of arcsin(x), etc. that lies in the range listed above. A math table I use refers to the principal value of the arcsine of y as Arcsin(y), using capitalization to indicate the distinction. I don't know how standard this is. The notation sin^-1(y) should always indicate the function (returning the principal value). The choice of range for the principal value can be a source of confusion. A student recently asked us about arccot, because his calculator used a range of -pi/2 to pi/2 for the principal value of the arccotangent, instead of 0 to pi as I listed above, while his textbook used the range I listed. I found that at least one respected Web source also used the range -pi/2 to pi/2. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
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