Trigonometrik tengsizliklar: matematika darsi

$\sin x\ge\dfrac{1}{2}$ tengsizlikning yechimi bitta oraliq emas, balki har $2\pi$ da takrorlanadigan cheksiz ko'p oraliq: $\left[\dfrac{\pi}{6}+2\pi n;\,\dfrac{5\pi}{6}+2\pi n\right]$ . Bunday oraliqni birlik aylanadan o'qish oson. Murakkab tengsizliklar esa almashtirish va formulalar bilan shu ko'rinishga keltiriladi.

Birlik aylanada yechish

Tengsizlikni tenglikka almashtirib, ikki chegaraviy nuqtani belgilaymiz.
Sinus uchun ordinata bo'yicha, kosinus uchun abssissa bo'yicha kerakli yoyni ajratamiz.
Yoyni soat strelkasiga teskari yo'nalishda yozamiz: chap uchi har doim o'ng uchidan kichik bo'lishi kerak.
Davrni qo'shamiz: sinus va kosinusda $2\pi n$ , tangens va kotangensda $\pi n$ .

\sin x\ge\dfrac{1}{2}

: ordinatasi

\dfrac{1}{2}

dan katta yoy,

\dfrac{\pi}{6}

dan

\dfrac{5\pi}{6}

gacha

\cos x\ge\dfrac{1}{2}

: yoy

-\dfrac{\pi}{3}

dan

\dfrac{\pi}{3}

gacha

Bu yerda abssissasi $\dfrac{1}{2}$ dan katta yoy olinadi. $\dfrac{5\pi}{3}$ dan $\dfrac{\pi}{3}$ gacha deb yozish xato bo'lar edi.

Sinus va kosinus uchun umumiy yechim

Quyida $n\in Z$ . Yechim: chap uchi $<x<$ o'ng uchi. Noqat'iy tengsizlikda ( $\ge$ , $\le$ ) oraliq uchlari ham kiradi.

Tengsizlik	Shart	Chap uchi	O'ng uchi
Tengsizlik $\sin x>a$	Shart $-1\le a<1$	Chap uchi $\arcsin a+2\pi n$	O'ng uchi $\pi-\arcsin a+2\pi n$
Tengsizlik $\sin x<a$	Shart $-1<a\le 1$	Chap uchi $-\pi-\arcsin a+2\pi n$	O'ng uchi $\arcsin a+2\pi n$
Tengsizlik $\cos x>a$	Shart $-1\le a<1$	Chap uchi $-\arccos a+2\pi n$	O'ng uchi $\arccos a+2\pi n$
Tengsizlik $\cos x<a$	Shart $-1<a\le 1$	Chap uchi $\arccos a+2\pi n$	O'ng uchi $2\pi-\arccos a+2\pi n$

Chegaradagi $a$ qiymatlarini alohida ko'ramiz, u yerda javob o'z-o'zidan ko'rinadi:

Tengsizlik	Yechimi	Sababi
Tengsizlik $\sin x>1$ , $\sin x>2$	Yechimiyechim yo'q	Sababisinus $1$ dan oshmaydi
Tengsizlik $\sin x\ge 1$	Yechimi $x=\dfrac{\pi}{2}+2\pi n$	Sababifaqat $\sin x=1$ qoladi
Tengsizlik $\sin x\le 1$ , $\sin x>-2$	Yechimiistalgan $x$	Sababisinus har doim shu chegarada
Tengsizlik $\sin x<1$	Yechimi $x\ne\dfrac{\pi}{2}+2\pi n$	Sababifaqat $\sin x=1$ chiqadi
Tengsizlik $\sin x\le -1$	Yechimi $x=-\dfrac{\pi}{2}+2\pi n$	Sababifaqat $\sin x=-1$ qoladi
Tengsizlik $\sin x>-1$	Yechimi $x\ne-\dfrac{\pi}{2}+2\pi n$	Sababifaqat $\sin x=-1$ chiqadi
Tengsizlik $\cos x\ge 1$	Yechimi $x=2\pi n$	Sababifaqat $\cos x=1$ qoladi
Tengsizlik $\cos x<1$	Yechimi $x\ne 2\pi n$	Sababifaqat $\cos x=1$ chiqadi
Tengsizlik $\cos x\le -1$	Yechimi $x=\pi+2\pi n$	Sababifaqat $\cos x=-1$ qoladi
Tengsizlik $\cos x>-1$	Yechimi $x\ne\pi+2\pi n$	Sababifaqat $\cos x=-1$ chiqadi

Tengsizlik	Yechimi
$\sin x>0$	$2\pi n<x<\pi+2\pi n$
$\sin x<0$	$-\pi+2\pi n<x<2\pi n$
$\cos x>0$	$-\dfrac{\pi}{2}+2\pi n<x<\dfrac{\pi}{2}+2\pi n$
$\cos x<0$	$\dfrac{\pi}{2}+2\pi n<x<\dfrac{3\pi}{2}+2\pi n$

Tangens va kotangens uchun umumiy yechim

Bu funksiyalar istalgan qiymatni qabul qiladi, shuning uchun $a$ ga cheklov yo'q. Davr $\pi$ . Tangens $\left(-\dfrac{\pi}{2};\,\dfrac{\pi}{2}\right)$ da o'sadi, kotangens $(0;\,\pi)$ da kamayadi.

Tengsizlik	Yechim	Uchlari
Tengsizlik $\operatorname{tg}x>a$	Yechim $\operatorname{arctg}a+\pi n<x<\dfrac{\pi}{2}+\pi n$	Uchlariikkalasi ochiq
Tengsizlik $\operatorname{tg}x\ge a$	Yechim $\operatorname{arctg}a+\pi n\le x<\dfrac{\pi}{2}+\pi n$	Uchlarichap uchi yopiq
Tengsizlik $\operatorname{tg}x<a$	Yechim $-\dfrac{\pi}{2}+\pi n<x<\operatorname{arctg}a+\pi n$	Uchlariikkalasi ochiq
Tengsizlik $\operatorname{tg}x\le a$	Yechim $-\dfrac{\pi}{2}+\pi n<x\le\operatorname{arctg}a+\pi n$	Uchlario'ng uchi yopiq
Tengsizlik $\operatorname{ctg}x>a$	Yechim $\pi n<x<\operatorname{arcctg}a+\pi n$	Uchlariikkalasi ochiq
Tengsizlik $\operatorname{ctg}x\ge a$	Yechim $\pi n<x\le\operatorname{arcctg}a+\pi n$	Uchlario'ng uchi yopiq
Tengsizlik $\operatorname{ctg}x<a$	Yechim $\operatorname{arcctg}a+\pi n<x<\pi+\pi n$	Uchlariikkalasi ochiq
Tengsizlik $\operatorname{ctg}x\le a$	Yechim $\operatorname{arcctg}a+\pi n\le x<\pi+\pi n$	Uchlarichap uchi yopiq

Tangens va kotangensda uzilish nuqtalari noqat'iy tengsizlikda ham kirmaydi, chunki u yerda funksiya mavjud emas: $\operatorname{tg}x\ge 1$ uchun o'ng uch $\dfrac{\pi}{2}+\pi n$ doimo ochiq.

Murakkab argument

$\sin(kx+b)>a$ da avval qavsni butun burchak deb yechamiz, keyin $x$ ni ajratamiz. $k$ manfiy bo'lsa, avval juft-toqlik bilan minusni chiqaramiz. Masalan, $\cos 3x\ge\dfrac{\sqrt{2}}{2}$ uchun:

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