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

\[\begin{align*}\frac{{dy}}{{dx}} & = \frac{{\displaystyle \frac{{dr}}{{d\theta }}\sin \theta + r\cos \theta }}{{\displaystyle \frac{{dr}}{{d\theta }}\cos\theta - r\sin \theta }}\\ & \\ & = \frac{{\left( {1 + \sin \left( \theta \right)} \right)\sin \theta + \left( {\theta - \cos \left( \theta \right)} \right)\cos \theta }}{{\left( {1 + \sin \left( \theta \right)} \right)cos\theta - \left( {\theta - \cos \left( \theta \right)} \right)\sin \theta }}\end{align*}\]

Bu biroz 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, \(\theta = \frac{{3\pi }}{4}\) da \(r\)ni ham baholashimiz kerak.

\[{\left. {\frac{{dy}}{{dx}}} \right|_{\theta = \,\frac{{3\pi }}{4}}} = 0.2843 \hspace{0.75in} {\left. r \right|_{\theta = \frac{{3\pi }}{4}}} = 3.0633\]

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

Va nihoyat, bizga \(\theta = \frac{{3\pi }}{4}\) 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,

\[x = r\cos \left( \theta \right) = 3.0633\cos \left( {\frac{{3\pi }}{4}} \right) = - 2.1661\hspace{0.25in}y = r\sin \left( \theta \right) = 3.0633\sin \left( {\frac{{3\pi }}{4}} \right) = 2.1661\]

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

Tangens chiziq u holda,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{y = 2.1661 + 0.2843\left( {x + 2.1661} \right) = 0.2843x + 2.7819}}\]