PolarCoordinates - 1-masala yechimi | MathModel.uz

9.6-bo'lim : Qutb koordinatalari

1. $\displaystyle \left( {2,\frac{\pi }{7}} \right)$ qutbli koordinatali nuqta uchun bir xil nuqta uchun uch xil koordinatalar to'plamini aniqlang, ularning barchasi burchaklari $\displaystyle \frac{\pi }{7}$ dan farq qiladi va $ - 2\pi \le \theta \le 2\pi $ oralig'ida joylashgan.

Yechim

Bu muammo, albatta, biz birlik doirasini qanchalik yaxshi tushunamiz. Bu erda nuqta va ba'zi burchaklarning qisqacha eskizi.

$A sketch of the point with polar coordinates $\left( {2,\frac{\pi }{7}} \right)$ and several angles that end at it. The solid red line runs from the origin out to the point in the first quadrant at an angle of $\frac{\pi }{7}$ and a dashed red line continues it into the third quadrant. Curved arrows show the angles $\frac{{8\pi }}{7}$ and $ - \frac{{6\pi }}{7}$ ending at the dashed line and $ - \frac{{13\pi }}{7}$ ending at the solid line.$

Muammo bayonotida ko'rsatilgan diapazonda joylashgan qattiq qizil chiziqda tugaydigan salbiy burchak oddiygina $\frac{\pi }{7} - 2\pi = - \frac{{13\pi }}{7}$ ekanligini ko'rishimiz mumkin.

Agar biz qattiq chiziqni uchinchi kvadrantga uzatsak (i.e. chiziqli qizil chiziq) keyin kesilgan qizil chiziq bilan tugaydigan musbat burchak $\frac{\pi }{7} + \pi = \frac{{8\pi }}{7}$. Xuddi shunday, kesilgan qizil chiziq bilan tugaydigan salbiy burchak $\frac{\pi }{7} - \pi = - \frac{{6\pi }}{7}$.

Ushbu burchaklar bilan qolgan uchta nuqtani olish juda oddiy bo'lishi kerak.

2-qadam

Birinchi "yangi" nuqta uchun biz nuqtani olish uchun qattiq qizil chiziqda tugaydigan salbiy burchakdan foydalanishimiz mumkin.

\[\require{bbox} \bbox[2pt,border:1px solid black]{{\left( {2, - \frac{{13\pi }}{7}} \right)}}\]

3-qadam

Qolgan fikrlar uchun eslaylikki, agar biz manfiy $r$ dan foydalansak, nuqtani olish uchun burchak tugagan joydan “orqaga” boramiz. Shunday qilib, agar biz $r = - 2$ dan foydalansak, chiziqli qizil chiziqda tugaydigan har qanday burchak nuqtaga borish uchun birinchi kvadrant 2 birligiga "orqaga" o'tadi.

Bu chiziqli qizil chiziqda tugaydigan ijobiy va salbiy burchaklardan foydalangan holda qolgan ikkita nuqtani beradi,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{\left( { - 2, - \frac{{6\pi }}{7}} \right)}}\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\require{bbox} \bbox[2pt,border:1px solid black]{{\left( { - 2,\frac{{8\pi }}{7}} \right)}}\]