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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|        \
|         \  ***********
|        ****           ****
|     ***   \               **
|   **       \                 **
|  *          \                  **
|**            \                   *
|*              \                  *
+-----------------+-----------------+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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