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Here's some examples:

http://www.tech.mtu.edu/~avsergue/EET2233/Lectures/CHAPTER2.pdf

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They are NOT "defined" that way at all. What made you think they are?Why trigonometric functions are defined for unit circle, here "why" refers to what made them to define it this way

They are defined by ratios such as opposite over hypotenuse, opposite over adjacent, etc of a right triangle. The use of a unit circle is purely pedagogical and is done because it makes the hypotenuse 1 and thus simplifies the calculations.

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Then how a right triange have a angle greater than 183°, how can you have the ratio of sides.They are NOT "defined" that way at all. What made you think they are?

They are defined by ratios such as opposite over hypotenuse, opposite over adjacent, etc of a right triangle. The use of a unit circle is purely pedagogical and is done because it makes the hypotenuse 1 and thus simplifies the calculations.

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Because you define the sides to have direction in cartesian co-ordinates. The right angle is at the origin. If the horizontal side goes to the left it is negative. And so forth. It is trivially simple. Have you actually studied this stuff at all?Then how a right triange have a angle greater than 183°, how can you have the ratio of sides.

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Trigonometric functions may be defined as ratios of the sides of right triangles in an elementary mathematics texts, but they ARE defined in terms of distances on the unit circle in intermediate and advanced texts.They are NOT "defined" that way at all. What made you think they are?

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OK, I had forgotten that. The point I think is more that such a definition isn't necessary but certainly I mis-spoke in saying that they are not defined that way. I should have said they don't HAVE to be defined that way.Trigonometric functions may be defined as ratios of the sides of right triangles in an elementary mathematics texts, but they ARE defined in terms of distances on the unit circle in intermediate and advanced texts.

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Have you? Defining trigonometric numbers in terms of the unit circle is much easier and is in many ways the only possible way to do it.Have you actually studied this stuff at all?

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That's pretty cool, I hadn't seen the definition of the secant and tangent using a unit circle before. Neat!Defining trigonometric numbers in terms of the unit circle is much easier and is in many ways the only possible way to do it.

https://en.wikipedia.org/wiki/Unit_circle

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For what trig function is it the only way to do it?Have you? Defining trigonometric numbers in terms of the unit circle is much easier and is in many ways the only possible way to do it.

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Yes they are easiar to represent by unit circle, but i want to know what made them to define it for angle more than 180° , they may have only defined it for a right angle, is there anciant application, where sin or cos of angle more than 180° was used.Have you? Defining trigonometric numbers in terms of the unit circle is much easier and is in many ways the only possible way to do it.

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It think what you mean is "Why is it useful to define the trigonometric functions for angles greater than 90 degrees or less than zero degrees ?".Yes they are easiar to represent by unit circle, but i want to know what made them to define it for angle more than 180° , they may have only defined it for a right angle,

Think about wave motion represented as ##V = r\ sin (\omega t)## where t is a time. It is desirable that ##sin(\omega t)## vary between + 1 and -1 so the wave will have a symmetrical shape. It is desirable that there be no bound on the argument ##\omega t##. A bound on ##\omega t## would put a limit on how long the process could continue in time.

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Yoo, correct, thanks this shows that they are usefull for angle more than 180°s, and they defined it for unit circle because it was easy to represent it that wayIt think what you mean is "Why is it useful to define the trigonometric functions for angles greater than 90 degrees or less than zero degrees ?".

Think about wave motion represented as ##V = r\ sin (\omega t)## where t is a time. It is desirable that ##sin(\omega t)## vary between + 1 and -1 so the wave will have a symmetrical shape. It is desirable that there be no bound on the argument ##\omega t##. A bound on ##\omega t## would put a limit on how long the process could continue in time.

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@micromass, I really am interested in the answer to this question. It seems very strange to me that there can be one, but I know that you know math so ...For what trig function is it the only way to do it?

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And it is defined for every trigonometric function@micromass, I really am interested in the answer to this question. It seems very strange to me that there can be one, but I know that you know math so ...

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I did not watch the video because it appears to simply show how trig functions are defined via a unit circle, which has nothing to do with my question.And it is defined for every trigonometric function

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Did you get your answer, i think you asked for what trig. Functions it is defined for unit circle, answer was given earliar that it is defined for every trigonometric function, but i think that this is not your question, can you explain your question briefly.@micromass, I really am interested in the answer to this question. It seems very strange to me that there can be one, but I know that you know math so ...

The unit circle definitions are much older. The origin of the word "sine" is "half", referring to half of a chord of the circle. The triangle definition is due to Rheticus.They are NOT "defined" that way at all. What made you think they are?

They are defined by ratios such as opposite over hypotenuse, opposite over adjacent, etc of a right triangle. The use of a unit circle is purely pedagogical and is done because it makes the hypotenuse 1 and thus simplifies the calculations.

Versine? Exsecent? Can't think of how you construct them with just a triangle.For what trig function is it the only way to do it?

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No, I most emphatically did NOT ask that.Did you get your answer, i think you asked for what trig. Functions it is defined for unit circle

Have you? Defining trigonometric numbers in terms of the unit circle is much easier and is in many ways the only possible way to do it.

@micromass still has not answered me, so no, I do not have my answer.For what trig function is it the only way to do it?

I don't see how you think that. The versine = 1 - adjacent/hypotenuse. What does that have to do with a unit circle?Versine? Exsecent? Can't think of how you construct them with just a triangle.

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extending trigonometric functions [for angles in a right-triangle] to circular functions [for any real number].

I wish I could find a more succinct article to show this.. but this will have to do for now...

http://amsi.org.au/ESA_Senior_Years/PDF/Trigonometryfunctions2d.pdf

And how do you do "one minus" in planar geometry without a compass?I don't see how you think that. The versine = 1 - adjacent/hypotenuse. What does that have to do with a unit circle?

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Who cares? I was pointing out the DEFINITION of versine, not describing how you might attack it geometrically. The question I have asked, and which has still not been answered, including by your response, is this: which trig functions can ONLY be defined using a unit circle? Micromass made the claim that there are such and since he knows more math than I do I'm open to the fact that I'm missing something but I don't see what.And how do you do "one minus" in planar geometry without a compass?

I used the word construct in post 20 for a reason. TheI was pointing out the DEFINITION of versine, not describing how you might attack it geometrically. The question I have asked, and which has still not been answered, including by your response, is this: which trig functions can ONLY be defined using a unit circle? Micromass made the claim that there are such and since he knows more math than I do I'm open to the fact that I'm missing something but I don't see what.

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