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Trigonometry In Simple Terms A request has been made on Wikipedia for this article to be deleted.
This request is being discussed to form a consensus whether this is, or could be, an article appropriate for Wikipedia. Please see this page's entry on the votes for deletion page for details. Also see possible outlets for removed articles. If you feel deletion is not justified by Wikipedia deletion policy you may vote against its deletion. Please do not remove this notice or blank this page while the question is being considered. However, you are welcome to continue editing this article and improve it, especially if you can address the concerns of those who believe the article should be deleted. This page is intended as a simplified introduction to trigonometry. (This article is not always correctly formulated in mathematical language.) Simple introduction If you are unfamiliar with angles, where they come from, and why they were actually required, maybe this will help. In principle all angles and trigonometric functions are defined on the unit circle. The term unit stands for '1' in mathematics. We can apply everything later to any scaled version (but more of that later on). Therefore all the needed functions can be derived from a triangle inscribed in the unit circle: it happens to be a right triangle. (TODO: Show picture here!) The center point of the unit circle will be set in a cartesian coordinate system, with its origin, the point (0, 0), at the center. Thus our circle will be divided into four sections, or quadrants (quad = 4). Quadrants are always counted counter-clockwise, as is the default rotation of angular velocity (omega). Now we inscribe a triangle in the first quadrant (that is, where the x- and y-axes are assigned positive values) and let one leg of the angle be on the x-axis and the other be parallel to the y-axis. (Just look at the illustration for clarification). Now we let the hypotenuse (which is always 1, the radius of our unit circle) rotate counter-clockwise. You will notice that a new triangle is formed as we move into a new quadrant, not only because the sum of a triangle's angles can not be beyond 180°, but also because there is no way on a two-dimensional plane to imagine otherwise. So much for the preliminaries; but what about angles, what are they all about? Angle-values simplified Imagine the angle to be nothing more than exactly the size of the triangle leg that resides on the x-axis (the cosine). So for any given triangle inscribed in the unit circle we would have a angle whose value is the distance of the triangle-leg on the x-axis. Although this would be possible in principle, it is much nicer to have a independent variable, let's call it phi, which does not change sign during the change from one quadrant into another and is easier to handle (that means it is not necessarily always a decimal number). !!Notice that all sizes and therefore angles in the triangle are mutually directly proportional. So for instance if the x-leg of the triangle is short the y-leg gets long. That is all nice and well, but how do we get the actual length then of the various legs of the triangle? By using translation tables, represented by a function (therefore arbitrary interpolation is possible) that can be composed by algorithms such as taylor. Those translation-table-functions (sometimes referred to as LUT, Look up tables) are well known to everyone and are known as sine, cosine and so on. (Whereas of course all the abovementioned latter ones can easily be calculated by using the sine and cosine). In fact in history when there weren't such nifty calculators available, printed sine and cosine tables had to be used, and for those who needed interpolated data of arbitrary accuracy - taylor was the choice of word. So how can I apply my knowledge now to a circle of any scale. Just multiply the scaling coefficient with the result of the trigonometric function (which is referring to the unit-circle). And this is also why , which is really nothing more than a veiled version of the pythagorean theorem: , whereas the , a peculiarity of most unit constructs. Now you also see why it is so comfortable to use all those mathematical unit-circles. Another way to interprete a angle-value would be: A angle is nothing more than a translated 'directed'-length into which the information of the actual quadrant is packed and the applied type of trigonometric function along with its sign determines the axis ('direction'). Thus something like the translation of a (x,y)-tuple into polar coordinates is a piece of cake. However due to the fact that information such as the actual quadrant is 'translated' from the sign of x and y into the angular value (a multitude of 90) calculations such as for instance the division in polar-form isn't equal to the steps taken in the non-polar form. Oh and watch out to set the right signs in regard to the number of quadrant in which your triangle is located. (But you'll figure that out easily by yourself). I hope the magic behind angles and trigonometric functions has disappeared entirely by now, and will let you enjoy a more in-depth study with the text underneath as your personal tutor.
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