Trigonometrik tenglamalar: matematika darsi

$\sin x=\dfrac{1}{2}$ tenglamaning ildizlari cheksiz ko'p, chunki sinus davriy. Shuning uchun javob seriya ko'rinishida yoziladi, kesma berilganda esa undan keraklilari tanlanadi. Har qanday trigonometrik tenglama oxirida eng oddiy tenglamalarga keladi, shuning uchun avval ularni yechamiz, keyin keltirish usullarini ko'ramiz.

Eng oddiy tenglamalar

Tenglama	Yechim	Sharti
Tenglama $\sin x=a$	Yechim $x=(-1)^n\arcsin a+\pi n$	Sharti $\|a\|\le 1$ ; $\|a\|>1$ da yechim yo'q
Tenglama $\cos x=a$	Yechim $x=\pm\arccos a+2\pi n$	Sharti $\|a\|\le 1$ ; $\|a\|>1$ da yechim yo'q
Tenglama $\operatorname{tg}x=a$	Yechim $x=\operatorname{arctg}a+\pi n$	Shartiistalgan $a$
Tenglama $\operatorname{ctg}x=a$	Yechim $x=\operatorname{arcctg}a+\pi n$	Shartiistalgan $a$

Hamma joyda $n\in Z$ . Sinus formulasi aslida ikkita seriyani bitta qatorga yozadi: juft $n$ da $x=\arcsin a+2\pi k$ , toq $n$ da $x=\pi-\arcsin a+2\pi k$ . Kesmadan ildiz tanlashda ikki seriyali yozuv qulayroq.

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

ildizlari:

\dfrac{\pi}{6}+2\pi n

\dfrac{5\pi}{6}+2\pi n

. Ordinata bo'yicha ikki nuqta

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

ildizlari:

\pm\dfrac{\pi}{3}+2\pi n

. Abssissa bo'yicha ikki nuqta

Xususiy holatlar

Quyidagilarda umumiy formula emas, soddaroq yozuv ishlatiladi. Ularda davr $\pi$ bo'lishi yoki $2\pi$ bo'lishiga e'tibor bering.

Tenglama	Yechim
$\sin x=0$	$x=\pi n$
$\sin x=1$	$x=\dfrac{\pi}{2}+2\pi n$
$\sin x=-1$	$x=-\dfrac{\pi}{2}+2\pi n$
$\cos x=0$	$x=\dfrac{\pi}{2}+\pi n$
$\cos x=1$	$x=2\pi n$
$\cos x=-1$	$x=\pi+2\pi n$
$\operatorname{tg}x=0$	$x=\pi n$
$\operatorname{tg}x=1$	$x=\dfrac{\pi}{4}+\pi n$
$\operatorname{tg}x=-1$	$x=-\dfrac{\pi}{4}+\pi n$
$\operatorname{ctg}x=0$	$x=\dfrac{\pi}{2}+\pi n$
$\operatorname{ctg}x=1$	$x=\dfrac{\pi}{4}+\pi n$
$\operatorname{ctg}x=-1$	$x=\dfrac{3\pi}{4}+\pi n$

O'ng tomon manfiy bo'lganda:

$\sin x=-a$ uchun $x=(-1)^{n+1}\arcsin a+\pi n$ .
$\cos x=-a$ uchun $x=\pm(\pi-\arccos a)+2\pi n$ .
$\operatorname{tg}x=-a$ uchun $x=-\operatorname{arctg}a+\pi n$ .
$\operatorname{ctg}x=-a$ uchun $x=\pi-\operatorname{arcctg}a+\pi n$ .

Jadvalda bo'lmagan qiymatda javob arkfunksiya bilan qoladi: $\operatorname{tg}x=2$ uchun $x=\operatorname{arctg}2+\pi n$ .

Argument murakkab bo'lganda

$\sin(kx+b)=a$ ko'rinishida avval $kx+b$ butun qavsni yechamiz, keyin $x$ ni topamiz. Bo'lganda davr ham bo'linadi. Masalan:

\cos\left(2x-\frac{\pi}{3}\right)=\frac{1}{2}

2x-\frac{\pi}{3}=\pm\frac{\pi}{3}+2\pi n

Demak,

x=\dfrac{\pi}{3}+\pi n

yoki

x=\pi n

Ikki funksiyaning tengligi

Tenglama	Yechim	Sharti
Tenglama $\sin A=\sin B$	Yechim $A=B+2\pi n$ yoki $A=\pi-B+2\pi n$	Shartihar qanday $A,\ B$
Tenglama $\cos A=\cos B$	Yechim $A=B+2\pi n$ yoki $A=-B+2\pi n$	Shartihar qanday $A,\ B$
Tenglama $\operatorname{tg}A=\operatorname{tg}B$	Yechim $A=B+\pi n$	Sharti $\cos A\ne 0$ , $\cos B\ne 0$
Tenglama $\operatorname{ctg}A=\operatorname{ctg}B$	Yechim $A=B+\pi n$	Sharti $\sin A\ne 0$ , $\sin B\ne 0$
Tenglama $\sin A=\cos B$	Yechim $\sin A=\sin\left(\dfrac{\pi}{2}-B\right)$ ga keltiramiz	Shartihar qanday $A,\ B$

Tangensli holatda shart muhim. $\operatorname{tg}3x=\operatorname{tg}x$ dan $3x=x+\pi n$ , ya'ni $x=\dfrac{\pi n}{2}$ . Lekin toq $n$ da $\operatorname{tg}x$ mavjud emas, shuning uchun javob faqat $x=\pi k$ .

Kvadrat tenglamaga keltirish

Ifodada bitta funksiya qatnashsa yoki ayniyat bilan bitta funksiyaga keltirilsa, uni $t$ deb belgilaymiz. Sinus va kosinus uchun $|t|\le 1$ shartni albatta tekshiramiz, tangens uchun bunday cheklov yo'q. Bitta funksiyaga keltirish yo'llari:

Ko'rinish	Almashtirish	Natija
Ko'rinish $\cos^2x$ va $\sin x$	Almashtirish $\cos^2x=1-\sin^2x$	Natija $\sin x$ ga nisbatan kvadrat
Ko'rinish $\sin^2x$ va $\cos x$	Almashtirish $\sin^2x=1-\cos^2x$	Natija $\cos x$ ga nisbatan kvadrat
Ko'rinish $\cos 2x$ va $\sin x$	Almashtirish $\cos 2x=1-2\sin^2x$	Natija $\sin x$ ga nisbatan kvadrat
Ko'rinish $\cos 2x$ va $\cos x$	Almashtirish $\cos 2x=2\cos^2x-1$	Natija $\cos x$ ga nisbatan kvadrat
Ko'rinish $\cos 4x$ va $\cos 2x$	Almashtirish $\cos 4x=2\cos^2 2x-1$	Natija $\cos 2x$ ga nisbatan kvadrat
Ko'rinish $\operatorname{tg}x$ va $\operatorname{ctg}x$	Almashtirish $\operatorname{ctg}x=\dfrac{1}{\operatorname{tg}x}$	Natija $\operatorname{tg}x$ ga nisbatan kasr
Ko'rinish $\dfrac{1}{\cos^2x}$ va $\operatorname{tg}x$	Almashtirish $\dfrac{1}{\cos^2x}=1+\operatorname{tg}^2x$	Natija $\operatorname{tg}x$ ga nisbatan kvadrat

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