How do we convert this into rectangular equation? $r=5\theta$
$\endgroup$2 Answers
$\begingroup$$r=x^2+y^2,\,\tan(\theta)=\frac{y}{x}$,so we have
$\frac{r}{5}=\theta,\tan(\frac{r}{5})=\frac{y}{x}$
the rectangle equation is $$\tan\left(\frac{\sqrt{x^2+y^2}}{5}\right)=\frac{y}{x}$$
$\endgroup$ 1 $\begingroup$I get this easily, check that if i am wrong: $r = a\theta$, so in terms of Cartesian coordinates… $$x = a\theta\cos\theta,\qquad y =a\theta\sin\theta.$$
We can also solve for $x$ and $y$ through simple algebraic manipulation
We know: $r^2=x ^2+ y^2$. Let's solve for $x$ first:
$$\begin{align*}
r &= a\theta\\
r^2&=a^2\theta^2\\
x^2+y^2&=a^2\theta^2\\
x^2&=a^2\theta^2-y^2
\end{align*}$$
Square both sides and substitute $r^2=x^2+ y^2$, $y=r\sin\theta$ (hence $y^2=r^2\sin^2\theta$), $x^2=a^2\theta^2-r^2\sin^2\theta$ (hence $x^2=a^2\theta^2- a^2\theta^2\sin^2\theta$) we get
$$x^2=a^2\theta^2(1- \sin^2\theta).$$
$y$ follows easily
