I Need help in this and how to convert rectangular coordinates to polar coordinates and vice versa. my teacher vaguely described how to graph also. Also finding other polar representations of any point.
for example: find an equivalent way of writing (2, pi/4)?
finding polar representations of (2,pi/3?
convert (-2, 5pi/4) to rectangular coordinates.
convert (0,2) to polar coordinates, and r=5costheta to rec. coordinates.
whats the rules of that if r is negative suppose to do?
Polar coordinates are of the form , where is the directed distance (i.e., it can be positive or negative) between the point and the origin, and is the angle measured counterclockwise from the polar axis to the radial line containing the point. For example, the point (-1, 0) in rectangular coordinates would be in polar since the distance from the origin is 1 and the angle it makes with the axis is .
Look at the graph.
Two points have been plotted. For now, look at . The means that the point is 3 units from the origin, and the means that the line intersecting the point and origin makes an angle of with the polar axis.
So how would we convert this to rectangular coordinates? Notice that I have drawn a perpendicular from the point to the axis. This gives us a right triangle. You can see that the two legs of the triangle are of length and (the distance from the - and -axis respectively), and the hypotenuse is . Thus, by the Pythagorean theorem, we have
Notice also that we can take the tangent of to get
These two formulas make it pretty easy to convert from rectangular to polar coordinates. What about from polar to rectangular? Well, taking the sine and cosine of the angle, we get
and
These four formulas should be all you need for conversions. For our point , we have and . Thus
and our point is in rectangular coordinates.
Now, notice that is not unique. For example, an angle of is really the same as . Also, notice that we can make negative, and as long as we flip the angle around, we get the same point (look at the other point on the graph). So, any point can be written as
or
When graphing, you can do things in two ways. Either convert the polar equations to rectangular ones, or just try to plot some points directly. For an example of a polar graph of a curve, this is a graph of a curve called a limaçon (its equation is of the form ).
Does that help?
Find another angle that is the same as .
See my post above.
Hint: multiply both sides by to get .
is the directed distance between the point and the origin. If is positive, it indicates that is units from the origin at the angle , while if is negative, it means that the point is units from the origin, at the angle opposite of . So .