wpp 11-12

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Unit N Concept 7: Derive the Unit Circle

These pictures show how the unit circle is derived from these special right triangles. As long as r=1, we can see that the points in the graph is the same as the unit circle showing that these special triangles are connected with the unit circle, which is why they are connected. As I derived the circle I can see that the angles match with that of right triangles, also as I derive the special triangles they are the same as from the unit circle.

So... about those conic sections

1. For the parabola the mathematical definition is any point on a parabola that is at equal distance from the focus, and the directrix. The definition of a parabola plays a role in the properties of the parabola because as the definition stated that the focus would be equal distance from the parabola and the directrix, gives the parabola the property that it has, such as if anything were to hit the parabola then it would go back towards the focus point. The reason the parabola has that property is because of that universal definition of parabolas. The definition of the parabola also affects on the shape because if the definition was invalid, then parabolas would either not exist and so we would not have satellites therefore more likely would not achieve the space era. 
2. The focus of the parabola affects the shape of the parabola depending on the number of the focus point such as if the focus point is farther away from the vertex then the parabola gets wider of fatter, but when the focus is closer to the vertex or the number gets smaller then the parabola is skinnier. By the law of Physics, if something were to hit a paraboliod, such as a transmitter, then the signals must always hit at the focus point. P is the distance between the focus and the directrix, so it makes sense that the farther the focus is the fatter and the closer it is, it's skinnier. Sometimes though P is a negative, so in that case the parabola will open down or left, while for p positive the parabola will go up or right. 

3. The properties of the parabola apply to real life because in a paraboloid shaped object, with the definition of a parabola being the same length, from focus to a point on a parabola and the directrix, we know that any part an object hits at a perfect paraboloid, it will accurately hit into the focus point, which is why we have satellites. With the paraboloid everyone can watch TV.
www.antenna-theory.com/antennas/reflectors/dish.php