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Lesson 2. Polar Coordinates.


The rectangular coordinate system in the plane is one in which a point is located by specifying its directed distnaces from two fixed lines, the x and y axes. Now consider a coordinate system in which a point is located by specifying its distance and direction from a fixed point. This system is essentially involved in saying that a town B is 30 miles northeast of another town A, where the distance(30 miles) and direction(northeast) of B from A are given instead of the distance east and the distance north.

Let O be a fixed point in a plane and let ray OA be a fixed ary in the plane through O. The fixed point is called the pole, and the ray OA is called the polar axis. It may be regarded as the positive half of the x axis of the rectangular coordinate system, when changeing from one system to the other.

Consider now any point P(other than O) in the plane. Its position can be given its directed distance OP from O and the measure of the angle that ray OP makes with ray OA. Then OP is called the radius vector of P and its denoted by the Greek letter r. the angle AOP is called the vectorial angle of P and is denoted by the Greek letter q. The two numbers r and q are called the polar coordinates of P and are written as the ordered pair (r,q).

In plotting a point whose coordinates (r,q) are given, r is the directed distance from the origin to the point. It is further agreed that if q is a positive number, the angle of q rad is measured from ray OA as the initial side in the counterclockwise direction. If q is negative, the angle is to be measured in the clockwise direction. (It may, of course, be specified that the number q is to be interpreted as the number of degrees instead of the number of radians in angle AOP.) It is also agreed to associate the origin with coordinates (0,q), where q is any real number.

Thus there is defined a coordinate system which associates a definite point of the plane wich every ordered pair of real numbers(r,q).



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