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Tangent Function and the Unit Circle

Date: 06/06/2002 at 06:43:05
From: Andy
Subject: Trigonometric Functions and the Unit Circle

Hi Dr. Math, 

I have a question regarding a response in your archive:

  Trigonometric Functions and the Unit Circle
  http://mathforum.org/library/drmath/view/53941.html 

In this response, it is shown that 

  tan(theta) = infinity when theta = pi/2

Then it is dicussed that as theta passes through pi/2 the picture 
flips and tan(theta) becomes large negative. What I can't work out is 
why the picture flips? Why can the tangent line not move to the other 
side of the circle thus making tan(pi/2+0.01) large positive?

Many thanks,
Andy


Date: 06/06/2002 at 08:29:17
From: Doctor Rick
Subject: Re: Trigonometric Functions and the Unit Circle

Hi, Andy.

Yes, we can draw the vertical tangent line on the left instead of the 
right (I think this is what you mean). But then the picture flips 
left-to-right instead of top-to-bottom, and the result turns out the 
same:

    P+
     |\
     | \
     |  \
     |   \
     |    \
     |     \
     |      \
     |       \
  opp|        \
     |         \  ***********
     |        ****           ****
     |     ***   \               **
     |   **       \                 **
     |  *          \                  **
     |**            \                   *
     |*              \                  *
     *     adj=-1     \ A                *
     +-----------------+-----------------+B
     *                                   *
      *                                 *
      **                                *
        *                             **
         **                         **
           ***                   ***
              ****           ****
                  ***********

The tangent is the ratio of the y coordinate of P to the x coordinate 
of P. The y coordinate is now a large *positive* number, and the x 
coordinate is -1, so y/x is a large *negative* number, just as before.

Have I cleared things up for you?

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


Date: 06/06/2002 at 12:06:52
From: Andy
Subject: Trigonometric Functions and the Unit Circle

Hello, 

Sorry to keep on about it. If I can refer you to a previous question,

  Demonstrating Sin, Cos, Tan on the Unit Circle
  http://mathforum.org/library/drmath/view/54104.html 

So here tan = FE, which is indeed positive. Now if we flip it to the 
left, then geometrically, FE is still positive as it is above the 
horizontal and thus a positive distance. I understand that the ratio 
is negative and so when pi/2 < theta < pi the geometric 
representation of tan seems to fall down and this is why I am 
confused.

Many thanks,
Andy


Date: 06/06/2002 at 14:30:20
From: Doctor Rick
Subject: Re: Trigonometric Functions and the Unit Circle

Hi, Andy.

The tangent function is not just FE, it is FE/OE. (Actually, it would 
be better to call it EF/OE.) Note what Dr. Jerry said near the bottom 
of the exchange you cite:

  tan(t) = side opposite/side adjacent = FE/1

The length OE is 1 in his figure. If you flip the figure left for 
right, then you must use the *signed* distance OE (positive to the 
right, negative to the left), just as you use the signed distance for 
EF (positive going up, negative going down). Thus, as I said, you can 
either look at the point in the second quadrant where the tangent is 
a positive number divided by -1, or in the fourth quadrant where the 
tangent is a negative number divided by 1. Either way it is negative.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Trigonometry

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