Deriving Basic (Circular) Trigonometric Functions

Mathematics, Physics
[caption id="attachment_18674" align="alignright" width="401"] Figure 1[/caption] Draw two intersecting lines in space, illustrated below. The mathematician will not want to leave this simple drawing without completely pointing out its features and labeling those features. We do so to begin our understanding of basic (circular) trigonometric functions. [caption id="attachment_18675" align="alignleft" width="387"] Figure 2[/caption] We label the point of intersection of course – P will do. But the intersection produces what looks like slices in a pie. The size of those slices of pie were determined by how the two lines intersected, how "wide apart" the lines are. We label these as angles α (alpha) and β (beta). Superimposing a Circle [caption id="attachment_18676" align="alignright" width="387"] Figure 3[/caption] The title of this paper is understanding basic circular trigonometric functions. So at this stage,…
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Introduction to Polar Coordinates

[caption id="attachment_25281" align="alignright" width="403"] Polar rose: r = 2 sin (4*θ)[/caption]Frequently used in analytical geometry is the standard 2-dimensional x, y coordinate system called the Cartesian coordinate system (named after famous mathematician, René Descartes). It's time to branch out to a different system, the polar coordinates system. In fact, there are any of a number of ways of locating points in 2-D space. Conversion from Cartesian Coordinates The polar coordinates system utilizes an angle and a radius. It is relatively simple to change from the x-y system to an r-θ system. Drawing a circle centered at the origin on an x-y plane and then drawing a right triangle with the radius of the circle equaling r, then by definition, the side adjacent to the angle divided by the hypotenuse (longest…
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