Tan 90

Date: 07/10/98 at 00:13:51
From: David Chung
Subject: Math

Why is tan 90 undefined?

Date: 07/10/98 at 03:25:38
From: Doctor Floor
Subject: Re: Math

Hi David,

Thank you for sending your question to Dr. Math!

To see why tan 90 is undefined, we should look at the way the tangent 
is defined. In a triangle it is defined in the following way:


                  C
               *  *
            *     *
         *        *
      *alpha      *
    A*************B

When ABC is rectangular with the right angle at B, then 
tan(alpha) = BC/AB. But from the triangle definition we can't say a 
thing about alpha = 90, because then there is no triangle anymore. 

To solve this definition problem, mathematicians switched to the 
unit-circle definition:

Create a circle centered at a point A with radius 1. From A draw the 
legs of your angle alpha. You can do this in such a way that one leg 
goes horizontally to the right, where it meets the circle in a point 
B. The second leg is is drawn in such a way that the angle is upwards 
for positive values and downwards for negative ones. Draw the line 
tangent (!) to the circle through B. The second leg and the tangent 
line meet in a point, say E. Now tan(alpha) is defined as the length 
of BE.

        

Now it is clear that if alpha = 90 there is no intersection of the 
second leg and the tangent line, so that tan 90 is not defined. 

If you have a math question again, please send it to Dr. Math!

Best regards,

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