Scetching the graph of a region in polar co-ords

This is the same equation as before and shows that the region is bounded by the circle centered at the origin with radius cos(£) and angle £ between 0 and pi/2 and 3pi/2 and 2pi. In summary, the given equation in polar coordinates is r<cos(£) and the region is bounded by a circle with radius cos(£) and angles between 0 and pi/2 and 3pi/2 and 2pi. Graphically, this represents the region enclosed by the circle including its boundary.
  • #1
franky2727
132
0
right my problem is express the region given by x^2+y^2<x in plane polar co-ords, hence sketch the region.
so i end up getting r<cos£ and £ lies between 0 and pi/2 and 3pi/2 and 2pi (if there's a better way or writing this please show me) i can show more working if required but i think that should suffice, the part I'm stuck on is the drawing of the graph, never been good at showing things graphically so could someone please scetch it out for me and work through what comes from where? thanks in advance
 
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  • #2
(x-a)^2 + (y-b)^2 = r^2 is the equation of a circle radius r centred at (a,b). So your equation will be a circle radius root x centred at the origin. The inequality indicates that you want the region enclosed by the circle INCLUDING the boundary.

Alternatively you can express x and y as rcos(£) and rsin(£) respectively and rewrite the equation as rcos^2() + rsin^2(£) <= r^2
 

1. What are polar coordinates and how are they used in graphing?

Polar coordinates are a way of describing the location of a point on a two-dimensional plane using a distance from the origin and an angle. In graphing, they are used to plot points on a polar coordinate system, which is based on a central point (the pole) and a ray (the polar axis).

2. How do I convert Cartesian coordinates to polar coordinates?

To convert Cartesian coordinates (x,y) to polar coordinates (r,θ), you can use the following formulas: r = √(x² + y²) and θ = tan⁻¹(y/x). This will give you the distance from the origin and the angle in radians.

3. What is the process for graphing a region in polar coordinates?

To graph a region in polar coordinates, you can start by identifying the equations or inequalities that define the boundaries of the region. Then, plot the points on a polar coordinate system and connect them to create the boundary of the region. Finally, shade in the area to represent the region.

4. How do I determine the symmetry of a graph in polar coordinates?

In polar coordinates, a graph can have symmetry with respect to the polar axis, the pole, or both. To determine symmetry with respect to the polar axis, replace θ with -θ in the equation and check if the resulting graph is the same. To determine symmetry with respect to the pole, replace r with -r and check if the resulting graph is the same.

5. Can negative values be used in polar coordinates?

Yes, negative values can be used in polar coordinates. The distance from the origin (r) can be negative if the point is in the opposite direction from the positive r-axis. The angle (θ) can also be negative if the point is in the clockwise direction from the positive polar axis.

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