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### Demonstrating Sin, Cos, Tan on the Unit Circle

```
Date: 12/30/98 at 02:45:08
From: Ally Miller
Subject: Demonstrating tan, csc, sec, or cot

I am trying to "build a trig function," so to speak. For instance, you
can demonstrate cos and sin with a pendulum or a ferris wheel by
plotting the various heights a specific point reaches as it goes around
its circular path.

But how can you demonstrate the other four trig functions? Would a
sundial work? I didn't think that you could demonstrate them since all
have infinite ranges. Is there any real life example of a tan, csc,
sec, or cot function?

Thanks.
```

```
Date: 12/30/98 at 08:18:51
From: Doctor Jerry
Subject: Re: Demonstrating tan, csc, sec, or cot

Hi Ally,

Suppose that you draw or construct a unit circle (radius 1). Starting
from the center, draw a horizontal line to the right, from the center O
of the circle to a point E on its circumference.  Now draw an arbitrary
angle t (but 0 < t < 90) by drawing a line from O to a point T on the
circumference and then extend OT to a point F directly above E.
Finally, draw a line starting from T, directly down to OE at point G,
so that TG is perpendicular to OE:

Then the distance FE is tan(t), OG is cos(t), and TG is sin(t).

One might argue that the other trig functions are just reciprocals of
these three and do not need to be explained in the same way.

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

```
Date: 12/31/98 at 03:07:05
Subject: Re: Demonstrating tan, csc, sec, or cot

Thank you very much for the reply, but I just have a quick question.
I understand what the diagram should look like, but can you explain to
me why this works? Also, how do I show how the tan function changes
over an interval of at least 80 degrees? Thanks!

-Ally
```

```
Date: 12/31/98 at 07:44:22
From: Doctor Jerry
Subject: Re: Demonstrating tan, csc, sec, or cot

Hi Ally,

I can explain why it works in the following way. First, in triangle
OFE, the angle at E is a right angle. So:

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

The lengths of OE and OT are 1 since it is a unit circle. In the
triangle OTG, the right angle is at G, so:

sin(t) = side opposite/hypotenuse = TG/OT = TG

Also:

cos(t) = side adjacent/hypotenuse = OG/OT = OG

This diagram will work as t varies from 0 to 80 or even 85  degrees,
although EF becomes quite long. Did you notice that EF is TANGENT to
the circle at E?

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

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