Polar Coordinates R

Introduction :

The polar coordinates R system is an option for rectangular system. In polar coordinate system, instead of a using (x, y) coordinates, a point is represented by (r, θ). Where r delineate the length of a straight line from the point to the origin and θ delineate the angle that straight line makes with the horizontal axis. The θ as the angular coordinate and r is generally referred to as the radial coordinate. From the origin the distance of a point P is consider by a point r (an arbitrary fixed point provided by the symbol Q).

Equations for Polar Coordinates R:

Consider θ =angle between the radial line from point P to Q and the given line “θ = 0”, a kind of positive axis for polar coordinates r system. Polar coordinates r are referred in terms of ordinary Cartesian coordinates through the transformations

x = r cos θ
y = r sin θ

Where r ≥0 0≤ θ < 2π.

From these relation we can see that the polar coordinates r of point P delineates the Relation x2 + y2 = r2 (cos2 θ + sin2 θ) ⇒ x2 + y2 = r2 (so that, as we indicated, P(x, y) point is on a circle of radius r centered at Q), other hand, we can find θ by calculating the equation

tan θ = y/x =⇒ θ = arctan (y/x),

for θ in the interval 0 ≤ θ < 2π.

Examples of Polar Coordinates R:

1) The following are typical “slices” in polar coordinates r (see the margin):

Radial slice = {(r, θ): θ = π/4, 1 ≤ r ≤ 2}

Radial slice = {(r, θ): θ = 3π/2, 0.5 ≤ r ≤ 0.8}

Circular slice = {(r, θ): r = 1.2, π/4≤ θ ≤ π/2}

Circular slice = {(r, θ): r = 3, 3π/4≤ θ ≤ π}

Now we can start describing regions using slices.

2) The ideas in Example 6 show that the circumference, C, of the circle x2 + y2 = R2 can be described by both in polar coordinates r.

C = {(r, θ): r = R, and 0 ≤ θ < 2π},

Along with the Cartesian description

C = {(x, y): |y| = R2 − x2, and − R ≤ x ≤ R}.