Q&A #4069

Arc sin and Arc cos pictures and domain

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From: Pat Ballew (for Teacher2Teacher Service)
Date: Jun 02, 2000 at 01:45:05
Subject: Re: Arc sin and Arc cos pictures and domain

The Arc in arc sine and arc cosine is related to the idea that we use the
arc length (in radial lengths instead of the angle in degrees) when we work
with radian measures.

If that is confusing, not to worry, you don't have to know why they call it
that to know how to work with them, and now we work with them in both angles
and radians.  I will begin with what it all means in angles since that is the
most common point for people new to these terms...

Hopefully you know a little about functions and their inverses.  An example
that will be useful are the functions that convert temperature in degrees f
to degrees C.
one says c = 5/9 (f-32)  (this converts  degrees Fahrenheit to degrees
centigrade.  The inverse of this formula will convert degrees c back to
degrees f... f= 9/5 f + 32

Now what inverses do, is undo each other... If I put 100 centigrade into the
formula for f I get 212 out,,, and if I put the 212 into the formula for c, I
get my 100 back.  This will work with any number starting in either
temperature formula, put the answer in the other, and your back where you
started...... OK?  now Arc sine and arc cosine are just inverses of sine and
cosine... I will do arc sine, and leave the other for you....

We begin with some basic identities... I will use d for degrees because the
little degree symbol is hard to print here... and for each known relation in
sine, I will write the equivalent statement in arcsine notation
  Sin (30 d) = 1/2     -----------  arcsine(1/2) = 30 d
  sin (45 d) = sqrt(2)/2 --------   arcsine(sqrt(2)/2) = 45 d
  sin (-30 d) = -1/2   ------------ arcsine (-1/2) = -30 d

One way to introduce the arcsine (x) is to read it as "the angle that has a
sine of x?"
Now everything is fine until we get to something like ...

   sin (150) = 1/2 and we think arcsine (1/2 ) = OOOhhhh we already have an
angle with a sine of 1/2, it was 30 degrees, and if arcsine is going to be a
function, then for every input, there must be one and only one output...

 So we have to restrict the output to only give us values from -90 d to 90
degrees.   The domain is of course restricted too, but it is for a different
reason.. The only numbers that we get out of the sin function are numbers
between -1 and 1, so the only numbers we can use as inputs (domain) for the
arcsine function are these same values...

That last statement is not exactly true if we work in radians and think of
imaginary lengths of radians, but that is not probably a worry for you at
this level......

I hope you can see by a similar argument that the domain of the arccosine
function will also be (-1,1).  The range is from 0d to 180 d because cosine
is negative in the 2nd quadrant but not in the fourth, so we have to make a
different choice for our range (and is was just a choice).

I hope that is clear, it is a lot better when done with pictures and such,
but we are not quite to that point...

Good luck.

 -Pat Ballew, for the Teacher2Teacher service

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