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### 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/
```
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