1.6-bo'lim : Trig tenglamalarni kalkulyator yordamida yechish, II qism
6. \(\displaystyle {\tan ^2}\left( {\frac{w}{4}} \right) = \tan \left( {\frac{w}{4}} \right) + 12\) ning barcha yechimlarini toping. Ishingizda kamida 4 kasrdan foydalaning.
Yechim
Bu tenglama bizdan hech qachon faktor qilish so'ralgan har qanday narsadan juda farq qilishi mumkin, ammo bu biz faktor qilishimiz mumkin bo'lgan narsadir. Avval quyidagilarni ko'paytuvchilarga ajratish haqida o'ylab ko'ring,
\[{x^2} = x + 12\hspace{0.25in} \to \hspace{0.25in} {x^2} - x - 12 = \left( {x - 4} \right)\left( {x + 3} \right) = 0\]Agar biz ushbu algebraik tenglamani faktorlarga ajrata olsak, berilgan tenglamani xuddi shu tarzda faktorlar qilishimiz mumkin.
\[\begin{align*}{\tan ^2}\left( {\frac{w}{4}} \right) & = \tan \left( {\frac{w}{4}} \right) + 12\\ {\tan ^2}\left( {\frac{w}{4}} \right) - \tan \left( {\frac{w}{4}} \right) - 12 & = 0\\ \left( {\tan \left( {\frac{w}{4}} \right) - 4} \right)\left( {\tan \left( {\frac{w}{4}} \right) + 3} \right) & = 0\end{align*}\]Endi bizda nolga teng bo'lgan ikkita omil mahsuloti bor va shuning uchun asosiy algebraik xususiyatlar bo'yicha biz ega bo'lishimiz kerakligini bilamiz,
\[\tan \left( {\frac{w}{4}} \right) - 4 = 0\hspace{0.25in}{\rm{OR}}\hspace{0.25in}\tan \left( {\frac{w}{4}} \right) + 3 = 0\]2-qadam
Bu tenglamalarning har biri oldingi bobda yechilgan tenglamalarga o'xshaydi. Shuning uchun, biz har biri uchun yechim jarayonini eslab qolishingiz mumkin deb o'ylaymiz va biz unchalik ko'p tafsilotlarni kiritmaymiz. Agar jarayonga ishonchingiz komil bo'lmasa, oldingi bo'limga qaytib, ushbu muammoni hal qilishda davom etishdan oldin ba'zi muammolar bilan shug'ullanishingiz kerak.
Biz birinchi tenglamadan boshlaymiz va olish uchun tangensni ajratamiz,
\[\tan \left( {\frac{w}{4}} \right) = 4\]Kalkulyatorimiz yordamida biz quyidagilarni olamiz:
\[\frac{w}{4} = {\tan ^{ - 1}}\left( 4 \right) = 1.3258\]Tangenslar ishtirokidagi tenglamalarni yechish haqidagi bilimlarimizdan bilamizki, \(\left[ {0,2\pi } \right]\) diapazonidagi ikkinchi burchak \(\pi + 1.3258 = 4.4674\) bo'ladi.
Birinchi tenglamaning barcha yechimlari u holda,
\[\begin{align*}\frac{w}{4} & = 1.3258 + 2\pi n & \hspace{0.25in}{\rm{OR}}\hspace{0.25in}\frac{w}{4} & = 4.4674 + 2\pi n & \hspace{0.25in} n = 0, \pm 1, \pm 2, \ldots \\w & = 5.3032 + 8\pi n & \hspace{0.25in}{\rm{OR}}\hspace{0.25in}w & = 17.8696 + 8\pi n & \hspace{0.25in} n = 0, \pm 1, \pm 2, \ldots \end{align*}\]Endi ikkinchi tenglamani yechamiz.
\[\tan \left( {\frac{w}{4}} \right) = - 3\hspace{0.25in}\hspace{0.25in} \to \hspace{0.25in}\hspace{0.25in}\frac{w}{4} = {\tan ^{ - 1}}\left( { - 3} \right) = - 1.2490\]Birlik doirasi haqidagi bilimlarimizdan biz ushbu burchakka mos keladigan musbat burchak \(2\pi - 1.2490 = 5.0342\) ekanligini ko'rishimiz mumkin. Bu burchaklardan ham foydalanish mumkin, lekin biz minus belgisini yo'qotmaslik uchun ijobiy burchakdan foydalanamiz. Shuningdek, \(\left[ {0,2\pi } \right]\) diapazonidagi ikkinchi burchak \(\pi + \left( { - 1.2490} \right) = 1.8926\).
Ikkinchi tenglamaning barcha yechimlari u holda,
\[\begin{align*}\frac{w}{4} & = 1.8926 + 2\pi n & \hspace{0.25in}{\rm{OR}}\hspace{0.25in}\frac{w}{4} & = 5.0342 + 2\pi n & \hspace{0.25in} n = 0, \pm 1, \pm 2, \ldots \\w & = 7.5704 + 8\pi n & \hspace{0.25in}{\rm{OR}}\hspace{0.25in}w & = 20.1368 + 8\pi n & \hspace{0.25in} n = 0, \pm 1, \pm 2, \ldots \end{align*}\]Bularning barchasini birlashtirib, quyidagi echimlar to'plamini beradi.
\[\require{bbox} \bbox[2pt,border:1px solid black]{{\begin{array}{rc}w & = 5.3032 + 8\pi n,\hspace{0.25in} & w & = 7.5704 + 8\pi n\\w & = 17.8696 + 8\pi n,\hspace{0.25in} & w & = 20.1368 + 8\pi n\end{array}} \hspace{0.25in} n = 0, \pm 1, \pm 2, \ldots }\]Agar interval berilgan bo'lsa, biz \(n\) qiymatlarini o'rnatishni davom ettiramiz va bu oraliqda qaysi echimlar to'g'ri kelishini aniqlashimiz mumkin. Bizga oraliq berilmagani uchun bu biz imkon qadar uzoqdir.
E'tibor bering, bu erda ishlatgan o'nli kasrlar miqdoriga qarab, javoblaringiz yaxlitlash xatosi tufayli ulardan biroz farq qilishi mumkin. Har qanday farqlar ozgina bo'lishi kerak va faqat 4 atrofida paydo bo'lishi kerakth o'nlik kasr yoki shunga o'xshash.