Calculus III Polar Coordinates The previous section introduced parametric equati

Question

Calculus III Polar Coordinates The previous section introduced parametric equations where both variables x, y depend on a third one. A new way to represent cartesian coordinates is polar coordinates. This typically de- scribes the motion of an object on a circular path Definition 1 A point (x,y) n the plane makes an angle 0 with the x-axis, and its directed distance rom the origin is r cos 6 sin 6 tan rcos 0 thus Using these formulas we can write the cartesian point (x,y) in polar coor- rsin 0 dinates (r 0) Example 1 The cartesian point (1 is in the fourth quadrant. We can computer V2 and the angle 0 or 6 T. The point can have many forms inpolar coordinates, eg. (V2 H), (v 2, E), etc. The point (-1, i) in polar coordinates is in the third quadrant. First we move up from the x-axis by angle but then we have to move in the opposite direction by distance -1 from the origin. The rcos cartesian coordinates are rsin Definition 2 A polar curve is a curve defined using polar coordinates: r f(0) Example 2 The curve r constant describes a circle centered at the origin. The curve 0 constant describes a ray (line). The curve r 2 sin describes a circle. To see this we multiply by r on both sides, thus r2 2rsin 1 and center (0,1) Definition 3 We can generalize the above example to see .r asin is a circle of radius and center (0,9) r acos 0 is a circle of radius 4 and center (5,0) Example 3 The curves r 1+ sin 6 and r 1 cos 0 are called cardioids because they are heart shaped. To see that, we need to make a table of values for (r,6) and plot these points.

Explanation / Answer

1) In Polar coordinates a point can be expressed in an infinite number of ways,

because we can add 2 or 4 or 6 to the angle and we get the same direction .

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