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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