Asosiy tarkibga o'tish

1.5-bo'lim : Trig tenglamalarni kalkulyator yordamida yechish, I qism

3. \(\displaystyle 3 = 6 - 11\sin \left( {\frac{t}{8}} \right)\) ning barcha yechimlarini toping. Ishingizda kamida 4 kasrdan foydalaning.

Yechim

Sinusni (bir koeffitsient bilan) tenglamaning bir tomonida ajratib olish,

\[\sin \left( {\frac{t}{8}} \right) = \frac{3}{{11}}\]
2-qadam

Birinchidan, kalkulyatorimiz yordamida biz buni ko'rishimiz mumkin,

\[\frac{t}{8} = {\sin ^{ - 1}}\left( {\frac{3}{{11}}} \right) = 0.2762\]

Endi biz bu muammoda sinus bilan shug‘ullanmoqdamiz va bilamizki, \(y\) o‘qi birlik aylanada sinusni ifodalaydi va shuning uchun biz \(\frac{3}{{11}}\) ning \(y\) koordinatasiga ega bo‘lgan burchaklarni qidiramiz. Bu shuni anglatadiki, biz birinchi burchakda (bu bizning kalkulyatorimiz bergan) va ikkinchi kvadrantda burchaklarga ega bo'lamiz. Mana bu holat uchun birlik doirasi.

A unit circle with the axes labeled with the angles 0 and \(2\pi = 6.2832\), \(\frac{\pi }{2} = 1.5708\), \(\pi = 3.1416\) and \(\frac{{3\pi }}{2} = 4.7124\). Two lines from the center meet the circle just above the \(x\)-axis, one in the first quadrant labeled 0.2762 and one in the second quadrant labeled 2.8654. A dashed horizontal line connects the two points, showing they have the same \(y\)-coordinate of \(\frac{3}{{11}}\).

Birlik doirasining simmetriyasidan biz \(\pi - 0.2762 = 2.8654\) ikkinchi burchak ekanligini ko'rishimiz mumkin.

3-qadam

Ushbu bo'lim uchun eslatmalar muhokamasidan shuni bilamizki, biz ushbu ikki burchakka ega bo'lgach, ularning har biriga "\( + \,2\pi n\) for \(n = 0, \pm 1, \pm 2, \ldots \)" ni qo'shish orqali barcha mumkin bo'lgan burchaklarni olishimiz mumkin.

Bu shuni anglatadiki, bizda bo'lishi kerak,

\[\frac{t}{8} = 0.2762 + 2\pi n\hspace{0.25in}{\mbox{OR }}\hspace{0.25in}\frac{t}{8} = 2.8654 + 2\pi n\hspace{0.25in}n = 0, \pm 1, \pm 2, \ldots \]

Nihoyat, tenglamaning barcha yechimlarini olish uchun ikkala tomonni 8 ga ko'paytirish kifoya.

\[\require{bbox} \bbox[2pt,border:1px solid black]{{t = 2.2096 + 16\pi n\hspace{0.25in}{\mbox{OR }}\hspace{0.25in}t = 22.9232 + 16\pi n\hspace{0.25in}n = 0, \pm 1, \pm 2, \ldots }}\]

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.