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    Hey, how would I go about sketching r^2=a^2 (cos2theta)? from -pi to pi?
    Where a is a positive constant? I drew it and it just looks like 4 flower petals, I said r= +- a^2 (cos 2theta), but aren't both of these the same?

    Also, I have to sketch r^2=a^2 (sin 2theta) from the first sketch...

    sin 2x=2sinxcosx
    and sin x = cos (x-pi/2), but I really don't know where to go. Any help would be good. Thanks.
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    When I sketch polar co-ordinates, I work out first what r is for theta=0. Then I work out whether r increases or decreases as theta increases, find the turning points/maxima+minima and mark them on. then join them up.

    For r^2=a^2 (cos2theta)

    r=a at theta=0. r then decreases to zero at theta = pi/4, keeps going to r=-a at theta=pi/2 etc. Just make sure you join them up in the right order. Your four petals sounds right.

    Do the same for sin 2x, though I think its a rotation of 45 degrees
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    I don't know what you've drawn....!
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    In attachment, just a bit more symmetrical :p:

    As I said, just try to figure out what values r will fluctuate between (in this case, a and -a) and where those values will occur (each half pi), then join them up, noting that the line must pass through (0,0) at pi/4, 3pi/4 etc
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    I was sketching it but surely cos theta can't take values of < 0, as r^2=-Ba^2, where you would be taking the value of a negative number... So I just got the 2 loops on the side at 0 and pi...
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    If it helps, you could write it as:

    r=a\sqrt{\cos {2\theta}}

    But it works out the same as saying that there are two values for r for each value of theta (the positive and negative square root)
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    But doesn't that mean that the expression is not defined between pi/4 and 3pi/4, and also -pi/4 and 3pi/4?
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    I thought it was -pi to +pi?

    Since cos repeats itself, you can define that expression between any two values of theta as long as they are at least 2pi different. I don't think I understand what you are asking

    Try sketching the curves like I have done, and check that the values satisfy the original equation
 
 
 
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