Secant, Cosecant, and Cotangent

Date: 01/16/97 at 18:20:41
From: Anonymous
Subject: Trig

Dear Dr. Math,

I'm an eleventh grader and I have a problem understanding equations 
using the secant, cosecant, and cotangent functions.  Is there any way  
you can help me?  The problems I have difficulty with are like this:

In the interval 0 <= x < 2pi, identify:

A. all values at which the function is undefined
B. all x-intercepts

I thank you very much,
Jim Hagler

Date: 01/24/97 at 14:51:13
From: Doctor Toby
Subject: Re: Trig

Here is how I remember these things:

First, remember the formulas for each function in terms of sine and 
cosine:

tan = sin/cos
cot = cos/sin
sec = 1/cos 
csc = 1/sin

If all you want to do now is solve problems A and B above, skip down 
to the row of asterisks below.  The stuff before that may help you if 
you have other problems along similar lines, but it's not necessary 
for problems A and B.

Memorize these facts about the sine and cosine (which perhaps you 
already have memorized):

  In the first quadrant (0 < x < pi/2), sine and cosine are both 
  positive.

  In the second quadrant (pi/2 < x < pi), only sine is positive.

  In the third quadrant (pi < x < 3 pi/2), neither sine nor cosine is 
  positive.

  In the fourth quadrant (3 pi/2 < x < 2 pi), only cosine is positive.

Remember the order: both; sine; neither; cosine.

Ultimately, these facts are true because the cosine is the 
x-coordinate and the sine is the y-coordinate of a point on a circle 
of radius 1 centred at the origin.

If you ever need to know the sign of one of the other trig functions,
just use the information you memorized about the sine and cosine.  For 
example, what sign is the cotangent in the second quadrant?  Cosine is 
negative there, while sine is positive there, so calculate:

  cot = cos/sin = -/+ = -; cotangent is negative there.

**********************************************************************

Sine and cosine are both continuous functions defined everywhere.
Therefore, whenever one of them changes sign, its value is zero 
there. So sine is zero at 0 and at pi, while cosine is zero at pi/2 
and at 3 pi/2.

The formulas for the other trig functions in terms of sine and cosine
give these functions as fractions. A fraction is undefined whenever 
its denominator is zero; otherwise, it is zero whenever its numerator 
is zero.  For example, cotangent = cosine/sine, so cotangent is 
undefined when sine is zero (at 0 and at pi) and zero when cosine is 
zero (at pi/2 and at 3 pi/2).  You can calculate this for the other 
functions the same way.

(Note that some functions might never be undefined or zero.)

-Doctor Toby,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/