# Prove if a polar function involves only the rational numbers and sin, cos, tan functions, it can be written in rectangular form.

Prove if a function only have including the rational numbers and sin, cos, tan function and $r$, you always could write it in rectangular form. Ex. For $r=2/(2+2\cos\theta)$ it could be represent in rectangular form. Rectangular form is where the $r$ in the original function is $\sqrt{x^2+y^2}$ and you could create a function in terms of $x$ and $y$ that is $=0$ that is refer to the original function

Also, another question would be: What other trig function could involved in a function if it could be represent in rectangular form.

is there a complete answer for this question?

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Could you define rectangular form? – Alex Becker Feb 28 '12 at 23:56
@Alex: Conversion from polar to rectangular coordinates. – Brian M. Scott Feb 29 '12 at 0:12
My tentative translation : "If a curve is described by a polar equation $r = f(\cos(\theta), \sin(\theta))$ (with $f \in \mathbb{Q}(X)$), can we define the same curve using a cartesian equation?" – Joel Cohen Feb 29 '12 at 0:13
What about something like: $r(\theta)=\sin(\tan(\theta))$? (that is, are you explicitly disallowing function composition?) – deoxygerbe Feb 29 '12 at 1:03
@deoxygerbe - yes, no composition – Victor Feb 29 '12 at 21:48

## 1 Answer

Substitute $r = \sqrt{x^2+y^2}$, $\cos(\theta) = x/\sqrt{x^2+y^2}$, $\sin(\theta) = y/\sqrt{x^2+y^2}$, $\tan(\theta) = y/x$ in the polar equation. Is that what you mean?

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yes, that is my meaning – Victor Feb 29 '12 at 0:37