Why are you allowed to do this?

Main Question or Discussion Point

So if I have something like this..

[tex]rcos\theta =-r^{2}sin^{2}\theta[/tex]

I can cancel out one of the r to get

[tex]cos\theta = rsin^{2}\theta[/tex]

but how come when you have something like..

[tex]sin^2\theta = sin\theta[/tex]

and say you are trying to find the zeros of this equation, you can't just do

[tex]sin\theta = 1[/tex]

Is it because in the first example, we assume that r never = 0 so you can cancel it out where as in the [tex]sin\theta [/tex] example, it could be 0? Thanks.
 

Answers and Replies

Hurkyl
Staff Emeritus
Science Advisor
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You are right -- you can never divide by an expression that may be zero, and cancelling is a form of division.

Incidentally, I would have said that both of those examples of cancelling are illegal. You can only do the first one if r is nonzero, but that is not generally true! It is, of course, legal whenever you do happen to know that r is nonzero -- for example, if you happen to split a problem into two cases, one where r is zero, and one where r is nonzero, then clearly in the second case, you'd be allowed to cancel an r.
 
Oh ok. Thanks for clearing that up! I forget that canceling is division! Silly me.
 
symbolipoint
Homework Helper
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Thanks for the fun example, motonoob101. You will have a quadratic equation with variable of cosine of theta:

[tex] \[
\begin{array}{l}
r\,\cos \theta + r^2 \,\sin ^2 \theta = 0 \\
r\,\cos \theta + r^2 \,(1 - \cos ^2 \theta ) = 0 \\
r\,\cos \theta + r^2 - r^2 \,\cos ^2 \theta = 0 \\
r^2 \cos ^2 \theta - r\,\cos \theta - r^2 = 0 \\
OR \\
\cos ^2 \theta - \frac{1}{r}\cos \theta - 1 = 0 \\
\end{array}
\]
[/tex]
 
246
1
also note that in you first example you "loose" a solution when deviding with r, namely r = 0, just like you loose solutions when deviding by sin in the second example.
 
HallsofIvy
Science Advisor
Homework Helper
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"lose", not "loose".

(I don't know why that irks me so much more than other misspellings! Perhaps because "loose" is a perfectly good word, just the wrong one.)
 

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