Answer to a Trig Question

Date: 1 May 1995 13:36:13 -0400
From: Jhone Ebert
Subject: Help me Dr. Math!

Hiya!
Well, we have a little math question for you.

What is the range of sin x = 1 / csc x ?

This was a test question, in my Algebra II / Trigonometry class.
My teacher marked it wrong when I answered All REAL numbers.  
She says that it is the answer is all real numbers except those that are 
integral multiples of pi.

But my argument is right because 1 / lim x -> 0+ is 1 / infinite which is 0.
Therefore all real numbers work.

I am 15 and my friend is 15 and we both think we are right.
What do you think?

Thanks,
James and Chonabot
jyukes@nsn.scs.unr.edu

From: Dr. Sydney
Subject: Re:  Help me Dr. Math!
Date: 3 May 1995 09:53:10 -0400

Hello again!

It seems that I misunderstood your question the first time I read and
answered it, so let me give it another shot.  I thought you were asking what
is the range of the function y = sin x (= 1/csc x), but some of the other
math doctors have told me that when you ask what the range of an equation
is, you are asking for what x values does the equation hold.  So, to find
the range of the equation

sin x = 1/csc x

we have to ask ourselves for what values of x does the equation work.  I
think your teacher has a point on this one, because csc x just is not defined
for integral multiples of pi.  You are right that 1/infinity is 0, and that
is a good point, but I think the question is probably really asking at what
values x are sin x and csc x nice...

It is kind of like asking for what values the following equation holds (What is
the range of this equation?)

x^2 - 1
-------  =  x + 1
 x - 1

Think about this one, and see if you can figure out an answer and see why it
is similar to your question.  (It has a different twist to it, but see if
you can figure it out!

Write back with any more problems!

--Sydney, "Dr. Math"