Keyinchalik, ushbu bo'limdagi eslatmalardan formuladan foydalanib, bizda,

\[\begin{align*}\frac{{dy}}{{dx}} & = \frac{{\displaystyle \frac{{dr}}{{d\theta }}\sin \theta + r\cos \theta }}{{\displaystyle \frac{{dr}}{{d\theta }}\cos\theta - r\sin \theta }}\\ & \\ & = \frac{{\left( {4\cos \left( {4\theta } \right)\cos \left( \theta \right) - \sin \left( {4\theta } \right)\sin \left( \theta \right)} \right)\sin \theta + \left( {\sin \left( {4\theta } \right)\cos \left( \theta \right)} \right)\cos \theta }}{{\left( {4\cos \left( {4\theta } \right)\cos \left( \theta \right) - \sin \left( {4\theta } \right)\sin \left( \theta \right)} \right)cos\theta - \left( {\sin \left( {4\theta } \right)\cos \left( \theta \right)} \right)\sin \theta }}\end{align*}\]

Bu juda chalkash lotin (ko'pincha shunday bo'ladi) va hech bo'lmaganda bu holatda biz qila oladigan juda ko'p soddalashtirish yo'q ...

Keyinchalik, biz oldingi bosqichdagi lotinni ham, \(\theta = \frac{\pi }{6}\) da \(r\)ni ham baholashimiz kerak.

\[{\left. {\frac{{dy}}{{dx}}} \right|_{\theta = \,\frac{\pi }{6}}} = \frac{1}{{3\sqrt 3 }} \hspace{0.75in} {\left. r \right|_{\theta = \frac{\pi }{6}}} = \frac{3}{4}\]

Ko'ryapsizmi, nega bizga bu ikkalasi ham kerak, to'g'rimi?

Va nihoyat, bizga \(\theta = \frac{\pi }{6}\) bo'lganda bo'ladigan \(x\) va \(y\) koordinatalari kerak. Ushbu qiymatlarni topish juda oson, chunki biz \(r\) nima ekanligini bilamiz va qutbni kartezian koordinatalariga aylantirish formulalarini ham bilamiz. Shunday qilib,

\[x = r\cos \left( \theta \right) = \frac{3}{4}\cos \left( {\frac{\pi }{6}} \right) = \frac{{3\sqrt 3 }}{8} \hspace{0.75in} y = r\sin \left( \theta \right) = \frac{3}{4}\sin \left( {\frac{\pi }{6}} \right) = \frac{3}{8}\]

Albatta, bizda tangens chizig'ining qiyaligi ham bor, chunki bu avvalgi bosqichda biz hisoblagan hosilaning qiymati.

Tangens chiziq u holda,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{y = \frac{3}{8} + \frac{1}{{3\sqrt 3 }}\left( {x - \frac{{3\sqrt 3 }}{8}} \right) = \frac{1}{{3\sqrt 3 }}x + \frac{1}{4}}}\]