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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
From: Meatball Head
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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