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/