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Math Help - [SOLVED] Polar graphs

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    Member >_<SHY_GUY>_<'s Avatar
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    Unhappy [SOLVED] Polar graphs

    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.
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    You must post a particular problem. Post a particular point to convert.
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    Member >_<SHY_GUY>_<'s Avatar
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    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?
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    Quote Originally Posted by >_<SHY_GUY>_< View Post
    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.
    Polar coordinates are of the form (r,\;\theta), where r is the directed distance (i.e., it can be positive or negative) between the point and the origin, and \theta 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 (1,\;\pi) in polar since the distance from the origin is 1 and the angle it makes with the axis is \pi.

    Look at the graph.



    Two points have been plotted. For now, look at \left(3,\;\frac{\pi}4\right). The r=3 means that the point is 3 units from the origin, and the \theta=\frac{\pi}4 means that the line intersecting the point and origin makes an angle of \frac{\pi}4 = 45^\circ 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 x and y (the distance from the y- and x-axis respectively), and the hypotenuse is r. Thus, by the Pythagorean theorem, we have

    r^2 = x^2 + y^2

    Notice also that we can take the tangent of \theta to get

    \tan\theta = \frac yx

    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

    \sin\theta = \frac yr\Rightarrow y = r\sin\theta

    and

    \cos\theta = \frac xr\Rightarrow x = r\cos\theta

    These four formulas should be all you need for conversions. For our point \left(3,\;\frac{\pi}4\right), we have r = 3 and \theta = \frac{\pi}4. Thus

    x = r\cos\theta = 3\cos\left(\frac{\pi}4\right) = \frac{3\sqrt2}2

    y = r\sin\theta = 3\sin\left(\frac{\pi}4\right) = \frac{3\sqrt2}2

    and our point is \left(\frac{3\sqrt2}2,\;\frac{3\sqrt2}2\right) in rectangular coordinates.

    Now, notice that \theta is not unique. For example, an angle of 15^\circ is really the same as 375^\circ. Also, notice that we can make r 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 (r,\;\theta),\;r\neq0 can be written as

    (r,\;\theta + 2n\pi),\;n\in\mathbb{Z}

    or

    \left(-r,\;\theta + (2n - 1)\pi\right),\;n\in\mathbb{Z}

    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 limašon (its equation is of the form r = a + b\cos\theta).



    Does that help?
    Attached Thumbnails Attached Thumbnails [SOLVED] Polar graphs-polar_coordinates.png   [SOLVED] Polar graphs-limacon.png  
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    Quote Originally Posted by >_<SHY_GUY>_< View Post
    for example: find an equivalent way of writing (2, pi/4)?
    Find another angle that is the same as \frac{\pi}4.

    Quote Originally Posted by >_<SHY_GUY>_< View Post
    finding polar representations of (2,pi/3?
    convert (-2, 5pi/4) to rectangular coordinates.
    See my post above.

    Quote Originally Posted by >_<SHY_GUY>_< View Post
    convert (0,2) to polar coordinates, and r=5costheta to rec. coordinates.
    Hint: multiply both sides by r to get r^2 = 5r\cos\theta = 5\left(r\cos\theta\right).

    Quote Originally Posted by >_<SHY_GUY>_< View Post
    whats the rules of that if r is negative suppose to do?
    r is the directed distance between the point and the origin. If r is positive, it indicates that (r,\;\theta) is r units from the origin at the angle \theta, while if r is negative, it means that the point is \lvert r\rvert = -r units from the origin, at the angle opposite of \theta. So (2,\;0^\circ) = (-2,\;180^\circ).
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