I have learned trigonometry, in school but never understood clearly what it is..just mugged up formula's and theorems to get clear the exams. But now i want to know what exactly, is trigonometry from the basics so that i can understand.
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A conceptual view of trigonometry is this: a simple diagram will convince you that if you know the side lengths of two sides of a triangle and the angle between them, then the length of the third side is uniquely determined (do a sketch and convince yourself of that!). A natural question then is:
If this angle that you are given happens to be a right angle, then you have Pythagoras. But what if you have an arbitrary angle? This question naturally leads to trigonometry.
Another way to arrive at the same thing: already the Greeks knew that if you start with a triangle and then scale it by a constant factor (zoom in or out, as it were), then the ratios of the corresponding side lengths don't change. In other words, these ratios only depend on the angles of the triangle (while, of course, each side length by itself cannot be computed just from the angles). So the question arises:
Yes - with trigonometry.
As for specific uses of sin, cos, and tan, they are just names for the aforementioned ratios of side lengths in the special case of right angled triangles. Now, you might be tempted to proceed and give names to these ratios in arbitrary triangles, but the great insight behind trigonometry is that you don't need to do that: you can express all those ratios using the already introduced notions of sin, cos, and tan. In other words, you can reduce all triangles to right angled ones (by using heights of the triangle, each of which is perpendicular to a side).
Also, things like Pythagoras generalise quite nicely. If you inspect the law of cosines, you will see that it is like "Pythagoras with an error term". So another way to express how one might arrive at trigonometry is this: you know how to express the hypotenuse in a right angled triangle. You also know that as you begin to shrink the angle while preserving the two side lengths, that hypotenuse gets shorter. By how much? You guessed it...
This is a good site(basic intro to trigonometry). Also Trigonometry by Gelfand is praised highly. It gives a more intuitive/theoretical approach to the subject (and it also has good problems/exercises).
In mathematics we encounter linear objects (i.e. cartesian coordinates, lengths, heights, distances, etc.) and circular type objects (i.e. angles, radii, rotations, etc). Often mathematical problems (and practical problems) require that we use both. Trigonometry is the set of mathematical tools by which we relate the linear and the circular objects. So sin, cos relate the angle and radius to x and y coordinates. Likewise, tan relates the x and y coordinates to the angle.