Matematikadagi do'konlar - formulalar, geometriya, trigonometriya

Matematikadagi aldash varaqalari to'plami.

Matematika xiyosi varaqlari - matematik belgilar

Geometriya belgilari

Belgi	Belgining nomi	Ma'nosi / ta'rifi	misol
∠	burchak	ikki nur bilan shakllangan	∠ABC \u003d 30 °
	o'lchov burchagi		ABC \u003d 30 °
	sharsimon burchak		AOB \u003d 30 °
∟	to'g'ri burchak	\u003d 90 °	a \u003d 90 °
°	daraja	1 ta aylanma \u003d 360 °	a \u003d 60 °
grad	daraja	1 ta aylanma \u003d 360 daraja	a \u003d 60 daraja
′	bosh Vazir	burchaklarning soni, 1 ° \u003d 60 '	a \u003d 60 ° 59 ''
″	ikki kishilik insult	ikkinchi bo'lim, 1 '\u003d 60 "	a \u003d 60 ° 59'59 "
	chiziq	cheksiz chiziq
Ab	chiziq segmenti	a nuqta dan b nuqtasi
	ray	a nuqta dan boshlanadigan chiziq
	yoy	aRC BUNI B nuqtasiga qadar	\u003d 60 °
⊥	perpendikulyar	perpendikulyar chiziqlar (90 °)	AC ⊥ BC
∥	parallel	parallel chiziqlar	AB ∥ CD
≅	mos keladi	geometrik shakllar va o'lchamlarning tengligi	Dohyz Doxyz
~	o'xshashlik	bir xil shakllar, turli o'lchamlar	DOBC ~ dexyz
Δ	uchburchak	uchburchakning shakli	Debcę dbcd
|  x —  u |	masofa	x va Y ballar orasidagi masofa	|  x —  u | \u003d 5
π	doimiy pi	π \u003d 3.141592654 ... aylananing uzunligi aylananing diametri bilan nisbati.	c. =  π ⋅  d. \u003d 2⋅ π ⋅  r
xursand	rentans	radiana burchakli birlik	360 ° \u003d 2p rad
c.	rentans	radiana burchakli birlik	360 ° \u003d 2p bilan
grad	gradikachilar / gononlar	burchak bloklari	360 ° \u003d 400 daraja
gina	gradikachilar / gononlar	burchak bloklari	360 ° \u003d 400 gina

Matematikadagi xaridorlar - geometriya formulalari

Matematikadagi xaridorlar - geometriyada formulalar:

• Aylana va uning qismlari uchun formulalar

Raqamli xususiyatlar	Rasm	Formula
Aylana maydoni		, qayerda R - doira radiusi, D. - aylananing diametri
Soha maydoni		, agar burchak o'lchami bo'lsa α radiolikesda ifodalangan
		, agar burchak o'lchami bo'lsa α darajalarda ifodalangan
Segmentning maydoni		, agar burchak o'lchami bo'lsa α radiolikesda ifodalangan
		, agar burchak o'lchami bo'lsa α darajalarda ifodalangan

Aylana va uning yoylari uchun formulalar

Raqamli xususiyatlar	Rasm	Formula
Aylanma		C \u003d2p R \u003dπ  D., qayerda R - doira radiusi, D. - aylananing diametri
Yoyning uzunligi		L.(α) = α R, agar burchak o'lchami bo'lsa α radiolikesda ifodalangan
		, agar burchak o'lchami bo'lsa α darajalarda ifodalangan

• Tegishli ko'pburchaklar

Ishlatilgan belgilar

Yon tomonda, perimetri va to'g'ri bo'lgan formulalar n. - Ugulnik

Qiymati	Rasm	Formula	Tavsif
Perimetr		P \u003d a	Perimetr bilan yon tomondan
Kvadrat			Erning ifodasi va yozuv ostida joylashgan doiralar radiusi orqali ifodalash
Kvadrat			Hududning yonidagi joyning ifodasi
Tomon			Yonning yozuvlari yozilgan doiralar orqali
Perimetr			Zerodlangan doiraning radiusi orqali perimetrning ifodasi
Kvadrat			O'rnatilgan doiraning radiusi orqali hududning ifodasi
Tomon			Tasvirlangan doiraning radiusi orqali yon tomonning ifodasi
Perimetr			Tasvirlangan doiraning radiusi orqali perimetrning ifodasi
Kvadrat			Tasvirlangan doiraning radiusi orqali hududning ifodasi

To'g'ri uchburchakning yon tomoni, perimetri va maydoni uchun formulalar

Qiymati	Rasm	Formula	Tavsif
Perimetr		P \u003d 3a	Perimetr bilan yon tomondan
Kvadrat			Hududning yonidagi joyning ifodasi
Kvadrat			Erning ifodasi va yozuv ostida joylashgan doiralar radiusi orqali ifodalash
Tomon			Yonning yozuvlari yozilgan doiralar orqali
Perimetr			Zerodlangan doiraning radiusi orqali perimetrning ifodasi
Kvadrat		Formula chiqishini ko'ring	O'rnatilgan doiraning radiusi orqali hududning ifodasi
Tomon			Tasvirlangan doiraning radiusi orqali yon tomonning ifodasi
Perimetr			Tasvirlangan doiraning radiusi orqali perimetrning ifodasi
Kvadrat			Tasvirlangan doiraning radiusi orqali hududning ifodasi

HEXAGAGON va To'g'ri oltietri va maydonining yon tomoni uchun formulalar

Qiymati	Rasm	Formula	Tavsif
Perimetr		P \u003d 6a	Perimetr bilan yon tomondan
Kvadrat			Hududning yonidagi joyning ifodasi
Kvadrat		S \u003d 3ar	Erning ifodasi va yozuv ostida joylashgan doiralar radiusi orqali ifodalash
Tomon			Yonning yozuvlari yozilgan doiralar orqali
Perimetr			Zerodlangan doiraning radiusi orqali perimetrning ifodasi
Kvadrat			O'rnatilgan doiraning radiusi orqali hududning ifodasi
Tomon		a \u003d r	Tasvirlangan doiraning radiusi orqali yon tomonning ifodasi
Perimetr		P \u003d 6R	Tasvirlangan doiraning radiusi orqali perimetrning ifodasi
Kvadrat			Tasvirlangan doiraning radiusi orqali hududning ifodasi

Yon tomonda, perimetri va kvadrat maydon uchun formulalar

Qiymati	Rasm	Formula	Tavsif
Perimetr		P \u003d 4a	Perimetr bilan yon tomondan
Kvadrat		S \u003da2	Hududning yonidagi joyning ifodasi
Tomon		a \u003d 2r	Yonning yozuvlari yozilgan doiralar orqali
Perimetr		P \u003d 8R	Zerodlangan doiraning radiusi orqali perimetrning ifodasi
Kvadrat		S \u003d4r2	O'rnatilgan doiraning radiusi orqali hududning ifodasi
Tomon			Tasvirlangan doiraning radiusi orqali yon tomonning ifodasi
Perimetr			Tasvirlangan doiraning radiusi orqali perimetrning ifodasi
Kvadrat		S \u003d2R2	Tasvirlangan doiraning radiusi orqali hududning ifodasi

• Uchburchak maydoni uchun formulalar

Rasm	Rasm	Maydonning formulasi	Belgilar
O'zboshimchalik bilan uchburchak			a - har qanday tomon r a - bu tomoni pastga tushirilgan baland
			a va b. - har qanday ikki tomon Dan - ular orasidagi burchak
		.	a, b, cPartiyalar, p. - yarim izohimetr Formula deyiladi "Formula Kironi"
			a - har qanday tomon B, s - qo'shni burchaklar
			a, b, c Partiyalar, r - Yaltirilgan doiraning radiusi, p. - yarim izohimetr
			a, b, c Partiyalar, R - tasvirlangan doiraning radiusi
		S \u003d2R2 gunoh A gunoh B. gunoh C.	A, b, c - burchaklar, R - tasvirlangan doiraning radiusi
Teng tomonli (to'g'ri) uchburchak			a yon tomon
			r - balandlik
			r - yozilgan doiralar radiusi
			R - tasvirlangan doiraning radiusi
O'ng uchburchak			a va b. - katetlar
			a - Katet, φ - qo'shni o'tkir burchak
			a - Katet, φ - qarama-qarshi o'tkir burchak
			c. - gipotenuse, φ - o'tkir burchaklar

• To'rtburchak joylar uchun formulalar

To'rtburchak	Rasm	Maydonning formulasi	Belgilar
To'rtburchaklar		S \u003d ab	a va b. - qo'shni tomonlar
			d.- diagonal, φ - diagonallar orasidagi to'rtta burchak
		S \u003d2R2 Gunt ph U yuqori formula almashtirishidan chiqadi D \u003d 2r	R - ta'riflangan doiraning radiusi, φ - diagonallar orasidagi to'rtta burchak
Parallelogramma		S \u003d a a	a - tomoni, r a - bu tomoni pastga tushirilgan baland
		S \u003d abgunt ph	a va b. - Qo'shni tomonlar, φ - ular orasidagi burchak
			d.1,  d.2 - diagonallar, φ - Ularning orasidagi to'rtta burchak
Kvadrat		S \u003d a2	a - kvadrat tomon
		S \u003d4r2	r - yozilgan doiralar radiusi
		Formula chiqishini ko'ring	d. - maydon diagonali
		S \u003d2R2 U yuqori formula almashtirishidan chiqadi d \u003d 2r	R - tasvirlangan doiraning radiusi
Romb		S \u003d a a	a - tomoni, r a - bu tomoni pastga tushirilgan baland
		S \u003da2 Gunt ph	a - tomoni, φ - rombning to'rt burchagidan birortasi
			d.1,  d.2 - diagonal
		S \u003d2araq Formula chiqishini ko'ring	a - tomoni, r - yozilgan doiralar radiusi
			r - Yaltirilgan doiraning radiusi, φ - rombning to'rt burchagidan birortasi
Trapezius			a va b. - asoslar, r - balandlik
		S \u003d m h	shodlik - o'rta chiziq, r - balandlik
			d.1,  d.2 - diagonallar, φ - Ularning orasidagi to'rtta burchak
			a va b. - asoslar, c. va d. - Yon tomonlar
Deltoli		S \u003d abgunt ph	a va b. - teng bo'lmagan tomonlar, φ - ular orasidagi burchak
			a va b. - teng bo'lmagan tomonlar, φ 1 - tomonlar orasidagi burchak teng a , φ 2 - tomonlar orasidagi burchak teng b..
		S \u003d(a + b) r	a va b. - teng bo'lmagan tomonlar, r - yozilgan doiralar radiusi
		Formula chiqishini ko'ring	d.1,  d.2 - diagonal
O'zboshimchalik bilan konveks to'rtburchagi			d.1,  d.2 - diagonallar, φ - Ularning orasidagi to'rtta burchak
Yozilgan to'rtburchaklar		,	a B C D - to'rtburchak tomonlarning uzunligi, p. - yarim izimerim, Formula deyiladi "Formula brahmamaslat"

• Muvofiqlashtirish usuli

Nuqtalar orasidagi masofa Lekin(x1; u1)  va DA(x2; u2)
Koordinatalar ( x;  u) Segmentning o'rtasi Ab tugaydi Lekin(x1;  u1va DA(x2;  u2)
Tenglama to'g'ridan-to'g'ri
Radiusi bilan dumaloq tenglama R va markazda markaz bilan ( x0;  u0)
Agar a Lekin ( x1;  u1va DA ( x2;  u2), keyin vektorning koordinatalari	(X2-X1; u2Qachonki1}
Vektorlar qo'shilishi	{x1; shilmoq1} +  {x2; shilmoq2} =  { xbiri  x2; shilmoqbiri  shilmoq2} {x1; shilmoq1}    {x2; shilmoq2} =  {xbiri  x2; shilmoqbiri  shilmoq2}
Vektorning ko'payishi {x; shilmoq} Raqamda k K.	k K.  {x; shilmoq} = k K. { k K.  x; k K.   shilmoq}
Vektorning uzunligi
Skarar vektorlar ishi va	∙    =  ∙ qayerda — vektorlar orasidagi burchak    va
Skarar vektorlar koordinatalarida ishlaydi	{x1; shilmoq1} va {x2; shilmoq2} ∙  = xbiri· x2 + shilmoqbiri· shilmoq2
Vektorning tarozilari {x; shilmoq}
Burchakning kosinasi vektorlar o'rtasida {x1; shilmoq1} va {x2; shilmoq2}
Vektorlarning perpencyulyulyulyulyulyulyulyityyati uchun zarur va etarli shart	{x1; shilmoq1} ┴  {x2; shilmoq2} ∙  = 0 yoki  xbiri· x2 + shilmoqbiri· shilmoq2= 0

Matematika xiyosi varaqlari - trigonometriya formulalari

Matematikadagi xaridorlar - trigonometriyadagi formulalar:

• Asosiy trigonometrik identifikatsiyalar

$$\mathrm{s.}\mathrm{men}{\mathrm{n.}}^{2}x+\mathrm{c.}o{\mathrm{s.}}^{2}x=1$$

$$t\mathrm{gina}x=\frac{\mathrm{s.}\mathrm{men}\mathrm{n.}x}{\mathrm{c.}o\mathrm{s.}x}$$

$$\mathrm{c.}t\mathrm{gina}x=\frac{\mathrm{c.}o\mathrm{s.}x}{\mathrm{s.}\mathrm{men}\mathrm{n.}x}$$

$$t\mathrm{gina}x\mathrm{c.}t\mathrm{gina}x=1$$

$$t{\mathrm{gina}}^{2}x+1=\frac{1}{\mathrm{c.}o{\mathrm{s.}}^{2}x}$$

$$\mathrm{c.}t{\mathrm{gina}}^{2}x+1=\frac{}{}$$

• Ikki marta argument (burchak)

$$\mathrm{s.}\mathrm{men}\mathrm{n.}2x=2\mathrm{c.}o\mathrm{s.}x\mathrm{s.}\mathrm{men}\mathrm{n.}x$$

$$\begin{array}{cc}\mathrm{s.}\mathrm{men}\mathrm{n.}2x& =\frac{2t\mathrm{gina}x}{1+t{\mathrm{gina}}^{2}x}\\ & =\frac{2\mathrm{c.}t\mathrm{gina}x}{1+\mathrm{c.}t{\mathrm{gina}}^{2}x}\\ & =\frac{2}{t\mathrm{gina}x+\mathrm{c.}t\mathrm{gina}x}\end{array}$$

$$\begin{array}{cc}\mathrm{c.}o\mathrm{s.}2x& ={\mathrm{kos}}^{2}x-\mathrm{s.}\mathrm{men}{\mathrm{n.}}^{2}x\\ & =2\mathrm{c.}o{\mathrm{s.}}^{2}x-1\\ & =1-2\mathrm{s.}\mathrm{men}{\mathrm{n.}}^{2}x\end{array}$$

$$\begin{array}{cc}\mathrm{c.}o\mathrm{s.}2x& =\frac{1-t{\mathrm{gina}}^{2}x}{1+t{\mathrm{gina}}^{2}x}\\ & =\frac{\mathrm{c.}t{\mathrm{gina}}^{2}x-1}{\mathrm{c.}t{\mathrm{gina}}^{2}x+1}\\ & =\frac{\mathrm{c.}t\mathrm{gina}x-t\mathrm{gina}x}{\mathrm{c.}t\mathrm{gina}x+t\mathrm{gina}x}\end{array}$$

$$\begin{array}{cc}t\mathrm{gina}2x& =\frac{2t\mathrm{gina}x}{1-t{\mathrm{gina}}^{2}x}\\ & =\frac{2\mathrm{c.}t\mathrm{gina}x}{\mathrm{c.}t{\mathrm{gina}}^{2}x-1}\\ & =\frac{2}{\mathrm{c.}t\mathrm{gina}x-t\mathrm{gina}x}\end{array}$$

$$\begin{array}{cc}\mathrm{c.}t\mathrm{gina}2x& =\frac{\mathrm{c.}t{\mathrm{gina}}^{2}x-1}{2\mathrm{c.}t\mathrm{gina}x}\\ & =\frac{2\mathrm{c.}t\mathrm{gina}x}{\mathrm{c.}t{\mathrm{gina}}^{2}x-1}\\ & =\frac{\mathrm{c.}t\mathrm{gina}x-t\mathrm{gina}x}{2}\end{array}$$

• Uchinchi argument formulalari (burchak)

$$\mathrm{s.}\mathrm{men}\mathrm{n.}3x=3\mathrm{s.}\mathrm{men}\mathrm{n.}x-4\mathrm{s.}\mathrm{men}{\mathrm{n.}}^{3}x$$

$$\mathrm{c.}o\mathrm{s.}3x=4\mathrm{c.}o{\mathrm{s.}}^{3}x-3\mathrm{c.}o\mathrm{s.}x$$

$$t\mathrm{gina}3x=\frac{3t\mathrm{gina}x-t{\mathrm{gina}}^{3}x}{1-3t{\mathrm{gina}}^{2}x}$$

$$\mathrm{c.}t\mathrm{gina}3x=\frac{\mathrm{c.}t{\mathrm{gina}}^{3}x-3\mathrm{c.}t\mathrm{gina}x}{3\mathrm{c.}t{\mathrm{gina}}^{2}x-1}$$

• Trigonometrik funktsiyalar yig'indisi formulalari

$$\mathrm{s.}\mathrm{men}\mathrm{n.}\alpha +\mathrm{s.}\mathrm{men}\mathrm{n.}\beta =2\mathrm{s.}\mathrm{men}\mathrm{n.}\frac{\alpha +\beta }{2}\cdot \mathrm{c.}o\mathrm{s.}\frac{\alpha -\beta }{2}$$

$$\mathrm{c.}o\mathrm{s.}\alpha +\mathrm{c.}o\mathrm{s.}\beta =2\mathrm{c.}o\mathrm{s.}\frac{\alpha +\beta }{2}\cdot \mathrm{c.}o\mathrm{s.}\frac{\alpha -\beta }{2}$$

$$t\mathrm{gina}\alpha +t\mathrm{gina}\beta =\frac{\mathrm{s.}\mathrm{men}\mathrm{n.}(\alpha +\beta )}{\mathrm{c.}o\mathrm{s.}\alpha \mathrm{c.}o\mathrm{s.}\beta }$$

$$\mathrm{c.}t\mathrm{gina}\alpha +\mathrm{c.}t\mathrm{gina}\beta =\frac{\mathrm{s.}\mathrm{men}\mathrm{n.}(\alpha +\beta )}{\mathrm{c.}o\mathrm{s.}\alpha \mathrm{c.}o\mathrm{s.}\beta }$$

$$(\mathrm{s.}\mathrm{men}\mathrm{n.}\alpha +\mathrm{c.}o\mathrm{s.}\alpha {)}^2=1+\mathrm{s.}\mathrm{men}\mathrm{n.}2\alpha$$

• Teskari trigonometrik funktsiyalar