Inverse Trigonometric Ratios

Date: 07/01/2003 at 09:27:03
From: Matthew
Subject: Inverse sin cos tan etc.

Why aren't the inverse trigonometric ratios equal to 1/(the ratio), 
as is the case for numbers?

e.g.
sin^-1 doesn't equal 1/sin (or cosec)
while x^-1 = 1/x

and why isn't then, say, sin^-1 equal to cosec?

This is the rule followed for all other numbers but it doesn't apply 
in this situation involving trigonometric ratios.

Date: 07/01/2003 at 16:07:36
From: Doctor Rob
Subject: Re: Inverse sin cos tan etc.

Thanks for writing to Ask Dr. Math, Matthew.

Yours is an excellent question. The answer involves the fact that the 
symbol __^(-1) is used for two different purposes, which are
confusingly related.

If c is a number, then c^(-1) is defined to be the reciprocal of c,
or 1/c.

If f(x) is a function, then f^(-1) is defined to be the inverse
function, that is, the function such that f[f^(-1)(x)] = x. In words, 
if you apply the inverse function to a value, then apply the function 
to that result, you get the original value back.

In your question, the trigonometric functions are functions, and so 
the second of these ideas applies.

The relation between these is as follows. If c is a nonzero number, 
there is a function f_c(x) defined by this: for all numbers x,

   f_c(x) = c*x

What is the inverse function of f_c?  It turns out that it is defined 
by this: for all numbers y,

   (f_c)^(-1)(y) = y/c = (1/c)*y = c^(-1)*y

and the inverse function involves the reciprocal of c.

Feel free to write again if I can help further.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/