Trig Function Domains

Date: 08/23/99 at 11:34:27
From: Nkiruka Ukachukwu
Subject: A sprig of trig

Which came first, trigonometry based on the simple triangle or 
trigonometry based on the triangle of the unit circle? I ask because 
not everything deduced from the trigonometry of the unit circle can be 
applied to a 'regular' triangle. For example, I always thought that 
the sine of an angle was "opposite over hypotenuse," but how do I take 
the sine of a 230-degree angle with a 'regular' triangle?

Is the unit circle system just that; merely a SYSTEM that is applied 
to other situations, but not something that can exist on its own like 
the natural number system can?

Date: 08/23/99 at 12:51:53
From: Doctor Rob
Subject: Re: A sprig of trig

Thanks for writing to Ask Dr. Math.

The simple triangle came first, then the unit circle later, as you 
seem to have guessed. The rule "opposite over hypotenuse" for the sine 
only applies to right triangles. Right triangles cannot have angles 
larger than 90 degrees. Furthermore, no triangle can have an angle of 
180 degrees or larger. The unit circle construction allows us to 
define trigonometric functions of any angle, not just the ones that 
appear in triangles.

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

Date: 08/23/99 at 22:48:17
From: Kiki Nwasokwa
Subject: Re: A sprig of trig

Were the trigonometric functions allowed by the unit circle made up or 
discovered?

Date: 08/24/99 at 11:53:32
From: Doctor Rob
Subject: Re: A sprig of trig

Thanks for writing back.

I think the question you are asking is whether mathematical ideas are 
invented or discovered. That is a very deep philosophical question, 
which has generated much debate over the years. Were circles invented 
or discovered?

Plato, the Greek philosopher, and those who support his position (the 
Platonists), believed that such abstract things exist independent of 
humanity, and are therefore are discovered, not invented. Their 
opponents believe that nothing abstract exists without human thought 
about it, and therefore they are invented by the mind of man, and have 
no independent existence.

I rather side with the Platonists on this one, but others are free to 
support either side.

If I have misinterpreted your question, do write back and explain more 
fully what it is you are asking.

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

Date: 08/24/99 at 17:48:27
From: Kiki Nwasokwa
Subject: Re: A sprig of trig

Why would we want to make up a system for defining trigonometric 
functions of angles larger than 180 degrees?

Date: 08/26/99 at 10:51:41
From: Doctor Rob
Subject: Re: A sprig of trig

For angles between 0 and 360 degrees, they are used in surveying. A 
polygon with more than 3 sides can have such angles as interior 
angles, and triangles have angles between 180 and 360 degrees as 
exterior angles.

When you study calculus, you will find that the trigonometric 
functions such as sin(x) can be defined for all real and complex 
values of x, and this function has very nice properties, some of which 
are not obvious at all. This extends the idea of these functions from 
their geometric origins to something much more general and much more 
abstract.

The sin(x) function is particularly useful in describing periodic 
behavior, such as a bouncing spring or a swinging pendulum, or even an 
alternating current in a wire.

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