Covariants

Date: 05/02/97 at 11:09:14
From: Vincent
Subject: Trigonometry

Dr. Math,

Please help me answer the following trigonometry problem:

Express tan(pi/2 + X) as a single function of X or as a constant.

Date: 05/02/97 at 11:45:34
From: Doctor Wilkinson
Subject: Re: Trigonometry

This problem involves the relations between the various trig functions 
and the "co" variants.  For example, sine and cosine, tangent and 
cotangent, secant and cosecant.  The "co" here relates to 
complementary angles.  The sine of an angle is the cosine of the 
complement of the angle, and so on. So we have the formulas:

 sin(pi/2 - x) = cos(x)
 tan(pi/2 - x) = cot(x)
 sec(pi/2 - x) = csc(x)

In your example, you have:

 tan(pi/2 + X)

We can apply the second formula with x = -X to get:

 tan(pi/2 + X) = cot(-X)

Now we are close.  The cot(-X) = -cot(X).  Remember the cotangent is
the quotient of the sine and the cosine, and we have the formulas:

 sin(-x) = -sin(x)
 cos(-x) =  cos(x)

So cot(-x) = cos(x)/(-sin(x)) = -cos(x)/sin(x) = -cot(x)

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