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Algebra

Trigonometrik funksiya qiymatlari

ortacha 220 daqiqa special anglesexact trig valuesunit circle30-60-90 triangle45-45-90 trianglesinecosinetangentcotangentsecantcosecantreference angle

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Maxsus burchaklarning aniq trigonometrik qiymatlari keyingi ayniyatlar, formulalar, trigonometrik tenglamalar, funksiyalar grafigi va geometriya masalalarida doimiy tayanch bo‘ladi. Maqsad jadvalni ko‘r-ko‘rona yodlash emas, qiymatlarni 30–60–90 va 45–45–90 uchburchaklari, birlik aylana koordinatalari va reference-angle mantiqidan qayta tiklay olishdir.

O'quv maqsadlari

  • 0°,30°,45°,60°,90° maxsus burchaklarni gradus va radianda moslashtirish
  • 45–45–90 uchburchak nisbatlarini chiqarish
  • 30–60–90 uchburchak nisbatlarini chiqarish
  • sin 30°, cos 30°, tan 30° aniq qiymatlarini topish
  • sin 45°, cos 45°, tan 45° aniq qiymatlarini topish
  • sin 60°, cos 60°, tan 60° aniq qiymatlarini topish
  • 0° va 90° quadrantal qiymatlarini unit circle’dan o‘qish
  • Birinchi chorak maxsus burchaklar jadvalini mantiqan tiklash
  • sin va cos qiymatlarining komplementar almashinuvini tushuntirish
  • tan qiymatini sin/cos nisbatidan hisoblash
  • cot qiymatini cos/sin yoki 1/tan orqali hisoblash
  • sec va csc ni reciprocal qiymatlar sifatida topish
  • Quadrant ishoralarini exact qiymatlarga to‘g‘ri qo‘llash
  • Reference angle orqali II chorak maxsus qiymatlarni topish
  • Reference angle orqali III chorak maxsus qiymatlarni topish
  • Reference angle orqali IV chorak maxsus qiymatlarni topish
  • Manfiy maxsus burchaklarning exact qiymatlarini topish
  • 360° dan katta maxsus burchaklarni reduksiya qilib exact qiymat topish
  • Radian ko‘rinishidagi maxsus burchaklarni exact baholash
  • Aniqlanmagan tan/sec/csc/cot holatlarini denominator orqali aniqlash
  • Exact radical qiymatlarni ratsionallashtirilgan ko‘rinishda yozish
  • Maxsus burchak qiymatlarini Topic66 ayniyatlari va keyingi trigonometrik masalalarga tayanch sifatida qo‘llash
Topic64’da burchak, quadrant, reference angle va birlik aylana koordinatalari qurildi. Endi shu geometriyadan maxsus burchaklarning aniq qiymatlarini olamiz. 45° uchun 45–45–90 uchburchak tomonlari 1:1:√2 nisbatda; radius 1 ga normallashtirilganda unit-circle koordinata (√2/2,√2/2) hosil bo‘ladi. 30° va 60° uchun 30–60–90 uchburchak tomonlari 1:√3:2 nisbatda; shundan 30° nuqta (√3/2,1/2), 60° nuqta (1/2,√3/2) keladi. Birinchi chorakdagi shu koordinatalar asosiy qiymatlarni beradi. Boshqa choraklarda modul reference angle’dan saqlanadi, ishora esa terminal nuqtaning x va y koordinatalaridan olinadi. Tangens y/x, cotangent x/y, secant 1/x va cosecant 1/y sifatida topiladi; denominator nol bo‘lgan joylarda tegishli funksiya aniqlanmaydi.

Ta'riflar

Maxsus burchak · Special angle

Trigonometrik qiymatlari oddiy exact sonlar yoki radikallar bilan ifodalanadigan standart burchak.

Asosiylari 0°,30°,45°,60°,90° va ularning unit-circle simmetriyalaridir.

Misol: π/6, π/4, π/3.

Bu emas: 17° odatda special angle emas.

💡 Reference angle special bo‘lsa katta burchak ham exact baholanadi.

Aniq trigonometrik qiymat · Exact trig value

Decimal approksimatsiyasiz, kasr va ildizlar bilan aynan ifodalangan trigonometrik qiymat.

Calculator rounded qiymat emas, algebraik exact natija.

Misol: sin30°=1/2.

Bu emas: sin30°≈0.5 ni yagona shakl deb olish.

💡 Exact qiymat keyingi algebraik soddalashtirish uchun qulay.

45–45–90 uchburchak

Ikki o‘tkir burchagi 45° bo‘lgan teng yonli to‘g‘ri burchakli uchburchak; tomonlar nisbati 1:1:√2.

45° exact qiymatlarini chiqaradi.

Misol: Katetlar 1,1 bo‘lsa gipotenuza √2.

Bu emas: 1:√3:2 nisbati.

💡 Unit radiusga bo‘lganda katetlar √2/2 bo‘ladi.

30–60–90 uchburchak

Burchaklari 30°,60°,90° bo‘lgan uchburchak; qarshi tomonlar nisbati 1:√3:2.

30° va 60° exact qiymatlarini beradi.

Misol: 30° qarshisidagi tomon gipotenuzaning yarmi.

Bu emas: 1:1:√2.

💡 Unit-circle koordinatalari shu nisbatdan olinadi.

Birinchi chorak maxsus jadvali

0°,30°,45°,60°,90° uchun sin va cos exact qiymatlari jamlanmasi.

Barcha boshqa special-angle qiymatlar uchun baza.

Misol: sin: 0,1/2,√2/2,√3/2,1.

Bu emas: Har bir quadrant uchun alohida jadval yodlash.

💡 Reference angle va ishora bilan qolgan choraklar tiklanadi.

Trigonometrik reciprocal · Reciprocal trig function

Asosiy trig funksiyaning teskari kasri sifatida aniqlangan funksiya.

sec=1/cos, csc=1/sin, cot=1/tan.

Misol: sec60°=2.

Bu emas: secθ=cos^{-1}θ ni inverse function deb talqin qilish.

💡 Bu yerda reciprocal va inverse-function tushunchalarini aralashtirmang.

Kotangens · Cotangent

sinθ≠0 bo‘lganda cotθ=cosθ/sinθ=1/tanθ.

Unit circle’da x/y nisbat.

Misol: cot30°=√3.

Bu emas: sinθ=0 da cotni 0 deyish.

💡 y=0 da undefined.

Sekans · Secant

cosθ≠0 bo‘lganda secθ=1/cosθ.

Cosine ning reciprocal qiymati.

Misol: sec60°=2.

Bu emas: cos90°=0 bo‘lsa sec90°=0.

💡 cos=0 da undefined.

Kosekans · Cosecant

sinθ≠0 bo‘lganda cscθ=1/sinθ.

Sine ning reciprocal qiymati.

Misol: csc30°=2.

Bu emas: sin0°=0 bo‘lsa csc0°=0.

💡 sin=0 da undefined.

Aniqlanmagan trig qiymat · Undefined trig value

Funksiya ta’rifidagi denominator nol bo‘lgani sabab real qiymat mavjud bo‘lmagan holat.

0 ga bo‘lish mumkin emas.

Misol: tan90°, sec90°, cot0°, csc0° undefined.

Bu emas: Qiymatni 0 deb yozish.

💡 Quadrantal burchaklarda ayniqsa tekshiriladi.

Ratsionallashtirilgan radikal

Denominatorida irrational ildiz qolmagan ekvivalent exact kasr.

1/√3 ni √3/3 ko‘rinishiga keltirish.

Misol: tan30°=√3/3.

Bu emas: tan30°=1/3.

💡 Har ikki ko‘rinish teng, ammo rationalized form ko‘p darsliklarda standart.

Komplementar burchaklar · Complementary angles

Yig‘indisi 90° yoki π/2 bo‘lgan ikki burchak.

30° va 60° bir-birining komplementi.

Misol: 30°+60°=90°.

Bu emas: 45° va 60°.

💡 Sin va cos qiymatlari almashadi.

Kofunksiya qiymati · Cofunction value

Komplementar burchaklarda sin va cos, tan va cot kabi juft funksiyalarning teng qiymatli bog‘lanishi.

sin30°=cos60°.

Misol: tan30°=cot60°.

Bu emas: sin30°=sin60°.

💡 Topic66’da identity sifatida chuqurlashadi.

Reference-angle transfer

Original burchak trig qiymatining modulini uning tayanch burchagi exact qiymatidan olish jarayoni.

Masalan 150° uchun reference angle 30°.

Misol: |sin150°|=sin30°=1/2.

Bu emas: sin150° ni sin150 sonidan calculator bilan boshlash.

💡 Ishora alohida quadrantdan olinadi.

Quadrant sign correction

Reference angle exact qiymatiga original quadrantga mos musbat yoki manfiy belgi qo‘yish.

Modul va sign ikki alohida qadam.

Misol: cos150°=−cos30°=−√3/2.

Bu emas: cos150°=+√3/2.

💡 Topic64 sign map’dan foydalanadi.

Quadrantal exact value

0°,90°,180°,270° va 360° kabi o‘q burchaklarida unit-circle koordinatadan bevosita olinadigan trig qiymat.

Reference triangle kerak emas.

Misol: sin270°=−1, cos270°=0.

Bu emas: tan90°=∞ ni real qiymat deyish.

💡 Undefined holat denominator orqali belgilanadi.

Maxsus unit-circle nuqta

Maxsus burchakka mos exact koordinatali P=(cosθ,sinθ) nuqta.

Koordinatadan barcha asosiy trig qiymatlar olinadi.

Misol: θ=π/3 → P=(1/2,√3/2).

Bu emas: P=(60°,30°).

💡 Burchak koordinata emas; cos/sin koordinatadir.

Exact tangent ratio

Maxsus burchakda tanθ exact sin/cos nisbatidan topiladi.

Alohida jadvalni yodlamasdan chiqarish mumkin.

Misol: tan60°=(√3/2)/(1/2)=√3.

Bu emas: tan60°=√3/2.

💡 cos=0 bo‘lsa ratio undefined.

Exact cotangent ratio

Maxsus burchakda cotθ exact cos/sin nisbatidan topiladi.

Tangentga reciprocal.

Misol: cot60°=1/√3=√3/3.

Bu emas: cot60°=√3.

💡 sin=0 bo‘lsa undefined.

Special-angle reduction

Katta yoki manfiy burchakni special reference angle’ga ega principal burchakka keltirish.

Koterminal va reference-angle qadamlar ketma-ketligi.

Misol: 750°→30°; sin750°=1/2.

Bu emas: 750° ni calculatorga kiritsagina exact deb hisoblash.

💡 Exact workflow periodiklikni ko‘rsatadi.

Exact radical simplification

Trig natijadagi ildiz va kasrlarni algebraik jihatdan standart ixcham ko‘rinishga keltirish.

√12/4 ni √3/2 ga qisqartirish kabi.

Misol: 2/√2=√2.

Bu emas: √2/2=√2 deb olish.

💡 Numerik qiymatni o‘zgartirmaslik kerak.

Approximate trig value

Maxsus bo‘lmagan burchakda calculator orqali olinadigan decimal yaqinlashuv.

Exact special qiymatlardan farqlanadi.

Misol: sin20°≈0.342.

Bu emas: sin30° ni faqat 0.500 deb berish.

💡 Mavzuda asosiy urg‘u exact qiymatlarga; approximation ikkilamchi.

Fundamental tushunchalar

45° qiymatini geometriyadan chiqarish

Katetlari 1,1 bo‘lgan isosceles right triangle gipotenuzasi √2; radius 1 ga normallashtirilganda har ikki koordinata √2/2.

$P_{45^\circ}=(\frac{\sqrt2}{2},\frac{\sqrt2}{2})$

sin45°=cos45°=√2/2.

30° qiymatini geometriyadan chiqarish

30–60–90 triangle nisbatidan 30° unit-circle nuqtasida y=1/2 va x=√3/2.

$P_{30^\circ}=(\frac{\sqrt3}{2},\frac12)$

sin30°=1/2, cos30°=√3/2.

60° qiymatini geometriyadan chiqarish

30° va 60° complementary bo‘lgani uchun koordinatalar almashadi.

$P_{60^\circ}=(\frac12,\frac{\sqrt3}{2})$

sin60°=√3/2, cos60°=1/2.

0° va 90° endpointlari

Unit circle o‘q nuqtalari maxsus jadvalning chetlarini beradi.

$P_0=(1,0),\quad P_{\pi/2}=(0,1)$

sin0=0,cos0=1; sin90=1,cos90=0.

Sine pattern

0°,30°,45°,60°,90° uchun sine qiymatlari √0/2,√1/2,√2/2,√3/2,√4/2 ketma-ketligiga mos.

$\sin:\frac{\sqrt0}{2},\frac{\sqrt1}{2},\frac{\sqrt2}{2},\frac{\sqrt3}{2},\frac{\sqrt4}{2}$

Qiymatlar 0 dan 1 gacha o‘sadi.

Cosine reverse pattern

Birinchi chorakda cosine sine patternining teskari tartibi.

$\cos:\frac{\sqrt4}{2},\frac{\sqrt3}{2},\frac{\sqrt2}{2},\frac{\sqrt1}{2},\frac{\sqrt0}{2}$

1 dan 0 gacha kamayadi.

Tangentni ratio orqali qurish

Tangent uchun alohida fundamental triangle yodlash shart emas; tan=sin/cos.

$\tan\theta=\frac{\sin\theta}{\cos\theta}$

tan0=0, tan30=√3/3, tan45=1, tan60=√3, tan90 undefined.

Cotangentni reciprocal orqali qurish

cot=1/tan=cos/sin; tan jadvalidan tez olinadi.

$\cot\theta=\frac{\cos\theta}{\sin\theta}$

cot0 undefined, cot30=√3, cot45=1, cot60=√3/3, cot90=0.

Secant exact qiymatlar

sec=1/cos; cosine nol bo‘lmasa reciprocal exact radical qiymat olinadi.

$\sec\theta=\frac1{\cos\theta}$

sec0=1, sec30=2√3/3, sec45=√2, sec60=2, sec90 undefined.

Cosecant exact qiymatlar

csc=1/sin; sine nol bo‘lmasa reciprocal exact qiymat olinadi.

$\csc\theta=\frac1{\sin\theta}$

csc0 undefined, csc30=2, csc45=√2, csc60=2√3/3, csc90=1.

Undefined-by-denominator gate

Har reciprocal/quotient trig funksiyada denominatorni avval tekshirish kerak.

$\cos\theta=0\Rightarrow\tan,\sec\text{ undefined};\quad \sin\theta=0\Rightarrow\cot,\csc\text{ undefined}$

Quadrantal angles eng ko‘p uchraydigan holat.

Reference angle exact transfer

Original burchak special reference angle’ga ega bo‘lsa modul birinchi chorak exact qiymatiga teng.

$|f(\theta)|=|f(\alpha)|$

Sign function va quadrantga bog‘liq.

QII exact transfer

II chorakda sin musbat, cos/tan manfiy; reference angle exact modul beradi.

$\sin(\pi-\alpha)=\sin\alpha,\quad \cos(\pi-\alpha)=-\cos\alpha$

Masalan 150°: sin=1/2, cos=−√3/2.

QIII exact transfer

III chorakda sin va cos manfiy, tan musbat.

$\sin(\pi+\alpha)=-\sin\alpha,\quad \cos(\pi+\alpha)=-\cos\alpha$

Masalan 225°: sin=cos=−√2/2, tan=1.

QIV exact transfer

IV chorakda sin/tan manfiy, cos musbat.

$\sin(2\pi-\alpha)=-\sin\alpha,\quad \cos(2\pi-\alpha)=\cos\alpha$

Masalan 330°: sin=−1/2, cos=√3/2.

Negative special angles

Manfiy angle clockwise rotation; coterminal reduction yoki x-axis symmetry exact qiymat beradi.

$\sin(-\alpha)=-\sin\alpha,\quad \cos(-\alpha)=\cos\alpha$

tan ham odd function.

Large-angle exact reduction

360° yoki 2π karralarini olib tashlab principal special angle topiladi.

$\theta\mapsto\theta-2\pi k$

Trig values coterminal angle’da bir xil.

Degree-radian dual recognition

π/6 ni 30° bilan, π/4 ni45°, π/3 ni60° bilan darhol bog‘lash exact evaluationni tezlashtiradi.

$30^\circ\leftrightarrow\frac\pi6,\ 45^\circ\leftrightarrow\frac\pi4,\ 60^\circ\leftrightarrow\frac\pi3$

0↔0, 90°↔π/2.

Cofunction swap

Complementary special anglesda sine va cosine qiymatlari almashadi; tangent va cotangent ham almashadi.

$\sin\theta=\cos(\frac\pi2-\theta),\quad\tan\theta=\cot(\frac\pi2-\theta)$

30°↔60°.

Radical rationalization

1/√3 kabi exact ratio denominatoridagi radicalni rationalize qilish mumkin.

$\frac1{\sqrt3}=\frac{\sqrt3}{3}$

Qiymat o‘zgarmaydi.

Exact vs decimal

Special burchaklarda √3/2 kabi exact qiymatni decimaldan ustun qo‘yish algebraik aniqlikni saqlaydi.

$\frac{\sqrt3}{2}\approx0.866025$

Approximation faqat talab qilinganda.

All-six-from-coordinate workflow

P=(x,y) exact unit-circle nuqta ma’lum bo‘lsa sin=y, cos=x, tan=y/x, cot=x/y, sec=1/x, csc=1/y.

$(\sin,\cos,\tan,\cot,\sec,\csc)$

Nol denominatorlar alohida tekshiriladi.

Reconstruction over memorization

Jadval esdan chiqsa 45–45–90 yoki 30–60–90 triangle’dan qiymatni qayta chiqarish mumkin.

$1:1:\sqrt2;\quad1:\sqrt3:2$

Bu yondashuv xatoni kamaytiradi.

Exact-value workflow

Inputni taning → principal/reference angle toping → first-quadrant exact modulni oling → quadrant sign qo‘ying → reciprocal/ratio kerak bo‘lsa hisoblang → undefined gate’ni tekshiring.

$\theta\to\alpha\to|f(\alpha)|\to\operatorname{sgn}\to f(\theta)$

Katta, manfiy va barcha quadrant special angles uchun bir xil workflow.

Formula kutubxonasi

45–45–90 nisbat

$$1:1:\sqrt2$$

45° exact qiymatlar generatori.

Shart: Teng yonli to‘g‘ri burchakli uchburchak.

Xususiy holatlar: Unit hypotenuse’da katetlar √2/2.

30–60–90 nisbat

$$1:\sqrt3:2$$

30° va 60° exact qiymatlar generatori.

Shart: 30°,60°,90° uchburchak.

Xususiy holatlar: Unit hypotenuse’da 1/2 va √3/2.

sin 0°

$$\sin0=0$$

Unit circle y-koordinata.

Xususiy holatlar: 0 rad.

cos 0°

$$\cos0=1$$

Unit circle x-koordinata.

Xususiy holatlar: P=(1,0).

sin 30°

$$\sin\frac\pi6=\frac12$$

30° sine exact value.

Xususiy holatlar: sin30°=1/2.

cos 30°

$$\cos\frac\pi6=\frac{\sqrt3}{2}$$

30° cosine exact value.

Xususiy holatlar: cos30°=√3/2.

tan 30°

$$\tan\frac\pi6=\frac{\sqrt3}{3}$$

30° tangent exact value.

Xususiy holatlar: Equivalent 1/√3.

cot 30°

$$\cot\frac\pi6=\sqrt3$$

30° cotangent exact value.

Xususiy holatlar: Reciprocal of tan30°.

sin 45°

$$\sin\frac\pi4=\frac{\sqrt2}{2}$$

45° sine exact value.

Xususiy holatlar: sin45°=cos45°.

cos 45°

$$\cos\frac\pi4=\frac{\sqrt2}{2}$$

45° cosine exact value.

Xususiy holatlar: Equal to sin45°.

tan 45°

$$\tan\frac\pi4=1$$

45° tangent exact value.

Xususiy holatlar: cot45°=1.

cot 45°

$$\cot\frac\pi4=1$$

45° cotangent exact value.

Xususiy holatlar: Self-reciprocal.

sin 60°

$$\sin\frac\pi3=\frac{\sqrt3}{2}$$

60° sine exact value.

Xususiy holatlar: sin60°=cos30°.

cos 60°

$$\cos\frac\pi3=\frac12$$

60° cosine exact value.

Xususiy holatlar: cos60°=sin30°.

tan 60°

$$\tan\frac\pi3=\sqrt3$$

60° tangent exact value.

Xususiy holatlar: cot30°=√3.

cot 60°

$$\cot\frac\pi3=\frac{\sqrt3}{3}$$

60° cotangent exact value.

Xususiy holatlar: Equivalent 1/√3.

sin 90°

$$\sin\frac\pi2=1$$

Top unit-circle point y-koordinata.

Xususiy holatlar: P=(0,1).

cos 90°

$$\cos\frac\pi2=0$$

Top unit-circle point x-koordinata.

Xususiy holatlar: Sec va tan undefined.

tan 90° undefined

$$\tan\frac\pi2\text{ aniqlanmagan}$$

0 ga bo‘lish sabab tangent mavjud emas.

Shart: cos(π/2)=0.

cot 90°

$$\cot\frac\pi2=0$$

cot=cos/sin=0.

Shart: sin(π/2)=1.

Secant reciprocal

$$\sec\theta=\frac1{\cos\theta}$$

Cosine reciprocal.

Shart: cosθ≠0.

Xususiy holatlar: sec60°=2.

Cosecant reciprocal

$$\csc\theta=\frac1{\sin\theta}$$

Sine reciprocal.

Shart: sinθ≠0.

Xususiy holatlar: csc30°=2.

Birinchi chorak sine pattern

$$(\sin0,\sin\frac\pi6,\sin\frac\pi4,\sin\frac\pi3,\sin\frac\pi2)=(0,\frac12,\frac{\sqrt2}{2},\frac{\sqrt3}{2},1)$$

Birinchi chorak maxsus sine qiymatlarining exact ketma-ketligi.

Birinchi chorak cosine pattern

$$(\cos0,\cos\frac\pi6,\cos\frac\pi4,\cos\frac\pi3,\cos\frac\pi2)=(1,\frac{\sqrt3}{2},\frac{\sqrt2}{2},\frac12,0)$$

Birinchi chorak maxsus cosine qiymatlarining exact ketma-ketligi.

Sine-cosine cofunction

$$\sin\theta=\cos(\frac\pi2-\theta)$$

Complementary anglesda sin/cos almashadi.

Xususiy holatlar: cosθ=sin(π/2−θ).

Tangent-cotangent cofunction

$$\tan\theta=\cot(\frac\pi2-\theta)$$

Complementary anglesda tan/cot almashadi.

Shart: Ikkala tomon aniqlangan.

QII special transfer

$$\sin(\pi-\alpha)=\sin\alpha,\quad \cos(\pi-\alpha)=-\cos\alpha$$

II chorak exact sign transfer.

Shart: 0<α<π/2.

QIII special transfer

$$\sin(\pi+\alpha)=-\sin\alpha,\quad \cos(\pi+\alpha)=-\cos\alpha$$

III chorak exact sign transfer.

Shart: 0<α<π/2.

QIV special transfer

$$\sin(2\pi-\alpha)=-\sin\alpha,\quad \cos(2\pi-\alpha)=\cos\alpha$$

IV chorak exact sign transfer.

Shart: 0<α<π/2.

Exact-value master workflow

$$f(\theta)=\operatorname{sgn}_{Q,f}\,|f(\alpha)|$$

Modul reference angle’dan, sign quadrantdan.

Shart: θ special reference angle α ga ega; f aniqlangan.

Xususiy holatlar: Undefined denominator gate oldin tekshiriladi.

Teoremalar va isbotlar

📐 45–45–90 exact-value teoremasi

Unit hypotenuse’li 45–45–90 uchburchakning har bir kateti √2/2 ga teng; shuning uchun sin45°=cos45°=√2/2 va tan45°=1.

Teng katetlar unit-circle diagonali bo‘ylab teng koordinata beradi.

Isbotni ko'rsatish

Berilgan: Katetlari teng bo‘lgan 45–45–90 uchburchak va unit hypotenuse.

Isbotlash kerak: Har katet √2/2 ekanini ko‘rsatish.

  1. Katetlarni 1 va 1 deb olsak gipotenuza √(1²+1²)=√2.
  2. Unit hypotenuse olish uchun barcha tomonlarni √2 ga bo‘lamiz.
  3. Katet 1/√2=√2/2 bo‘ladi.
  4. Shuning uchun unit-circle nuqta (√2/2,√2/2).

Demak sin45°=cos45°=√2/2 va tan45°=1.

📐 30–60–90 exact-value teoremasi

Unit hypotenuse’li 30–60–90 uchburchakda 30° qarshisi 1/2, 60° qarshisi √3/2; shu sabab 30° va 60° sine/cosine exact qiymatlari hosil bo‘ladi.

Equilateral triangle’ni ikkiga bo‘lish maxsus nisbatni beradi.

Isbotni ko'rsatish

Berilgan: Tomoni 2 bo‘lgan teng tomonli uchburchak.

Isbotlash kerak: 30–60–90 nisbat 1:√3:2 ni chiqarish.

  1. Balandlik asosni 1 va 1 ga bo‘ladi, gipotenuza 2.
  2. Pythagorean theorem: h²+1²=2².
  3. h²=3, h=√3.
  4. Shu bilan tomonlar 1,√3,2.

Unit hypotenuse’da 1/2 va √3/2 koordinatalar kelib chiqadi.

📐 Birinchi chorak maxsus jadval teoremasi

0°,30°,45°,60°,90° uchun unit-circle x,y koordinatalari maxsus triangle nisbatlari va o‘q endpointlaridan to‘liq aniqlanadi.

Faqat ikki special triangle va ikki endpoint butun jadvalni generatsiya qiladi.

Isbotni ko'rsatish

Berilgan: Unit circle va 0°,30°,45°,60°,90°.

Isbotlash kerak: Birinchi chorak sin/cos jadvalini chiqarish.

  1. 0° nuqta (1,0), 90° nuqta (0,1).
  2. 30° nuqta (√3/2,1/2).
  3. 45° nuqta (√2/2,√2/2).
  4. 60° nuqta (1/2,√3/2).
  5. x=cosθ, y=sinθ deb o‘qiymiz.

Birinchi chorak exact sin/cos jadvali to‘liq hosil bo‘ladi.

📐 Reference-angle exact transfer teoremasi

Agar θ ning reference angle’i α maxsus burchak bo‘lsa, |sinθ|=sinα, |cosθ|=cosα va aniqlangan bo‘lsa |tanθ|=tanα; sign original quadrantdan olinadi.

Akslantirishlar koordinata modullarini saqlaydi, faqat ishorani o‘zgartiradi.

Isbotni ko'rsatish

Berilgan: θ burchak va uning maxsus reference angle’i α.

Isbotlash kerak: Trig qiymat moduli α bilan bir xil, sign quadrantdan kelishini ko‘rsatish.

  1. Reference angle birinchi chorakdagi mos acute radius bilan bir xil x,y modullarini beradi.
  2. Y-axis reflection x ishorasini, x-axis reflection y ishorasini, 180° rotation ikkala ishorani o‘zgartiradi.
  3. Shuning uchun |sinθ|=sinα va |cosθ|=cosα.
  4. tan=sin/cos bo‘lgani uchun uning moduli ham tanα, sign nisbatdan keladi.

Barcha quadrant special values first-quadrant exact modul + sign orqali olinadi.

📐 Reciprocal-value teoremasi

Asosiy exact sin/cos/tan qiymat ma’lum bo‘lsa va denominator nol bo‘lmasa, sec=1/cos, csc=1/sin, cot=1/tan orqali reciprocal exact qiymatlar yagona aniqlanadi; denominator nol bo‘lsa funksiya aniqlanmagan.

Reciprocal funksiyalar yangi geometriya emas, mavjud ratio’larning teskari kasridir.

Isbotni ko'rsatish

Berilgan: Exact sinθ, cosθ va tanθ qiymatlari ma’lum.

Isbotlash kerak: Reciprocal trig qiymatlar va undefined holatlarni chiqarish.

  1. secθ=1/cosθ, shuning uchun cosθ≠0 bo‘lsa exact reciprocal olinadi.
  2. cscθ=1/sinθ, shuning uchun sinθ≠0 bo‘lsa exact reciprocal olinadi.
  3. cotθ=1/tanθ=cosθ/sinθ, shuning uchun sinθ≠0 bo‘lishi kerak.
  4. Denominator 0 bo‘lsa real reciprocal mavjud emas.

Asosiy exact jadval six-function jadvalni aniqlaydi va undefined nuqtalarni ko‘rsatadi.

Yechilgan misollar

oson sin30° ni exact toping.

💡 Maslahat: 30–60–90 triangle yoki jadvaldan foydalaning.

  1. 30° uchun unit-circle y-koordinata $1/2$.

✅ Javob: $\sin30^\circ=\frac12$

Nega bu usul ishlaydi: Sine terminal nuqtaning y-koordinatasi; 30–60–90 triangle qisqa katet/gipotenuza nisbatini beradi.

oson cos30° ni exact toping.

💡 Maslahat: 30° nuqtaning x-koordinatasini oling.

  1. 30° uchun $P=(\sqrt3/2,1/2)$.

✅ Javob: $\cos30^\circ=\frac{\sqrt3}{2}$

Nega bu usul ishlaydi: Cosine unit-circle x-koordinata.

oson tan30° ni exact toping.

💡 Maslahat: tan=sin/cos.

  1. $\tan30^\circ=(1/2)/(\sqrt3/2)=1/\sqrt3$
  2. Denominatorni rationalize qilamiz: $1/\sqrt3=\sqrt3/3$.

✅ Javob: $\tan30^\circ=\frac{\sqrt3}{3}$

Nega bu usul ishlaydi: Tangent exact sin/cos nisbatidan olinadi.

Muqobil usul: 30–60–90 triangle’da opposite/adjacent=1/√3.

oson cot30° ni exact toping.

💡 Maslahat: cot=1/tan yoki cos/sin.

  1. $\cot30^\circ=(\sqrt3/2)/(1/2)=\sqrt3$

✅ Javob: $\cot30^\circ=\sqrt3$

Nega bu usul ishlaydi: Cotangent tangent reciprocalidir.

oson sin45° ni exact toping.

💡 Maslahat: 45–45–90 triangle’dan foydalaning.

  1. Unit hypotenuse’da har katet $\sqrt2/2$.

✅ Javob: $\sin45^\circ=\frac{\sqrt2}{2}$

Nega bu usul ishlaydi: 45° triangle’da ikki katet teng.

oson cos45° ni exact toping.

💡 Maslahat: 45° da x va y koordinatalar teng.

  1. $P_{45^\circ}=(\sqrt2/2,\sqrt2/2)$

✅ Javob: $\cos45^\circ=\frac{\sqrt2}{2}$

Nega bu usul ishlaydi: Cosine x-koordinata; 45° diagonal simmetriyasi x=y beradi.

oson tan45° ni exact toping.

💡 Maslahat: sin45°/cos45°.

  1. $\tan45^\circ=(\sqrt2/2)/(\sqrt2/2)=1$

✅ Javob: $\tan45^\circ=1$

Nega bu usul ishlaydi: Teng katetlar ratio 1 beradi.

oson sec45° ni exact toping.

💡 Maslahat: sec=1/cos.

  1. $\sec45^\circ=1/(\sqrt2/2)=2/\sqrt2=\sqrt2$

✅ Javob: $\sec45^\circ=\sqrt2$

Nega bu usul ishlaydi: Secant cosine reciprocalidir.

oson sin60° ni exact toping.

💡 Maslahat: sin60°=cos30°.

  1. 60° unit-circle y-koordinata $\sqrt3/2$.

✅ Javob: $\sin60^\circ=\frac{\sqrt3}{2}$

Nega bu usul ishlaydi: 30° va 60° complementary bo‘lib, sin/cos qiymatlar almashadi.

oson cos60° ni exact toping.

💡 Maslahat: cos60°=sin30°.

  1. 60° unit-circle x-koordinata $1/2$.

✅ Javob: $\cos60^\circ=\frac12$

Nega bu usul ishlaydi: 30–60–90 triangle unit hypotenuse’da short leg 1/2.

oson tan60° ni exact toping.

💡 Maslahat: tan=sin/cos.

  1. $\tan60^\circ=(\sqrt3/2)/(1/2)=\sqrt3$

✅ Javob: $\tan60^\circ=\sqrt3$

Nega bu usul ishlaydi: Tangent exact coordinate ratio.

oson csc60° ni exact toping.

💡 Maslahat: csc=1/sin.

  1. $\csc60^\circ=1/(\sqrt3/2)=2/\sqrt3$
  2. Rationalize: $2/\sqrt3=2\sqrt3/3$.

✅ Javob: $\csc60^\circ=\frac{2\sqrt3}{3}$

Nega bu usul ishlaydi: Cosecant sine reciprocalidir.

oson sin0° va cos0° ni toping.

💡 Maslahat: Unit circle right endpointni o‘qing.

  1. 0° terminal nuqta $P=(1,0)$.
  2. Shuning uchun $\cos0^\circ=1$, $\sin0^\circ=0$.

✅ Javob: $\sin0^\circ=0,\quad\cos0^\circ=1$

Nega bu usul ishlaydi: Unit-circle coordinates bevosita exact qiymatlarni beradi.

oson sin90° va cos90° ni toping.

💡 Maslahat: Unit circle top endpointni o‘qing.

  1. 90° terminal nuqta $P=(0,1)$.

✅ Javob: $\sin90^\circ=1,\quad\cos90^\circ=0$

Nega bu usul ishlaydi: Top endpoint x=0,y=1.

oson tan90° va sec90° mavjudmi?

💡 Maslahat: Ikkalasining denominatorida cos90° bor.

  1. $\cos90^\circ=0$
  2. $\tan90^\circ=\sin90^\circ/\cos90^\circ=1/0$
  3. $\sec90^\circ=1/\cos90^\circ=1/0$

✅ Javob: Ikkalasi ham aniqlanmagan.

Nega bu usul ishlaydi: 0 ga bo‘lish real sonlarda aniqlanmagan.

⚠️ Undefined qiymatni ∞ yoki 0 deb yozmang.

ortacha sin120° ni exact toping.

💡 Maslahat: Reference angle 60°, QII’da sine musbat.

  1. $\alpha=180^\circ-120^\circ=60^\circ$
  2. QII’da sine musbat.
  3. $\sin120^\circ=\sin60^\circ=\sqrt3/2$

✅ Javob: $\sin120^\circ=\frac{\sqrt3}{2}$

Nega bu usul ishlaydi: Reference angle modulni, quadrant ishorani beradi.

ortacha cos120° ni exact toping.

💡 Maslahat: Reference angle 60°, QII’da cosine manfiy.

  1. $\alpha=60^\circ$
  2. $|\cos120^\circ|=\cos60^\circ=1/2$
  3. QII sabab manfiy.

✅ Javob: $\cos120^\circ=-\frac12$

Nega bu usul ishlaydi: Y-axis reflection x-coordinate ishorasini o‘zgartiradi.

ortacha tan120° ni exact toping.

💡 Maslahat: QII, reference angle 60°.

  1. $|\tan120^\circ|=\tan60^\circ=\sqrt3$
  2. QII’da tangent manfiy.

✅ Javob: $\tan120^\circ=-\sqrt3$

Nega bu usul ishlaydi: QII’da sin+/cos−, ratio manfiy.

ortacha sin135° ni exact toping.

💡 Maslahat: Reference angle 45°.

  1. $\alpha=180^\circ-135^\circ=45^\circ$
  2. QII’da sine musbat.

✅ Javob: $\sin135^\circ=\frac{\sqrt2}{2}$

Nega bu usul ishlaydi: Reference-angle transfer.

ortacha cos135° ni exact toping.

💡 Maslahat: QII cosine manfiy.

  1. $|\cos135^\circ|=\cos45^\circ=\sqrt2/2$
  2. QII sign: negative.

✅ Javob: $\cos135^\circ=-\frac{\sqrt2}{2}$

Nega bu usul ishlaydi: Unit-circle x-coordinate QII’da manfiy.

ortacha tan135° ni exact toping.

💡 Maslahat: Reference angle 45°, QII tangent manfiy.

  1. $|\tan135^\circ|=1$
  2. QII sign negative.

✅ Javob: $\tan135^\circ=-1$

Nega bu usul ishlaydi: tan=sin/cos va QII’da signs opposite.

ortacha sin150° ni exact toping.

💡 Maslahat: Reference angle 30°.

  1. $\alpha=30^\circ$
  2. QII’da sine musbat.

✅ Javob: $\sin150^\circ=\frac12$

Nega bu usul ishlaydi: Reference-angle module 1/2 saqlanadi.

ortacha cos150° ni exact toping.

💡 Maslahat: Reference angle 30°, QII cosine manfiy.

  1. $|\cos150^\circ|=\sqrt3/2$
  2. QII sabab minus.

✅ Javob: $\cos150^\circ=-\frac{\sqrt3}{2}$

Nega bu usul ishlaydi: Reference-angle transfer with quadrant sign.

ortacha tan150° ni exact toping.

💡 Maslahat: Reference angle 30°, QII tangent manfiy.

  1. $|\tan150^\circ|=\sqrt3/3$
  2. QII sign negative.

✅ Javob: $\tan150^\circ=-\frac{\sqrt3}{3}$

Nega bu usul ishlaydi: Tangent sign sin/cos ratio’dan keladi.

ortacha sin210° ni exact toping.

💡 Maslahat: Reference angle 30°, QIII sine manfiy.

  1. $\alpha=210^\circ-180^\circ=30^\circ$
  2. QIII’da sine manfiy.

✅ Javob: $\sin210^\circ=-\frac12$

Nega bu usul ishlaydi: 180° rotation y ishorasini o‘zgartiradi.

ortacha cos225° ni exact toping.

💡 Maslahat: Reference angle 45°, QIII cosine manfiy.

  1. $\alpha=45^\circ$
  2. $|\cos225^\circ|=\sqrt2/2$
  3. QIII sign negative.

✅ Javob: $\cos225^\circ=-\frac{\sqrt2}{2}$

Nega bu usul ishlaydi: QIII x-coordinate negative.

ortacha tan240° ni exact toping.

💡 Maslahat: Reference angle 60°, QIII tangent musbat.

  1. $\alpha=60^\circ$
  2. $|\tan240^\circ|=\sqrt3$
  3. QIII’da sin va cos ikkalasi manfiy, ratio musbat.

✅ Javob: $\tan240^\circ=\sqrt3$

Nega bu usul ishlaydi: Reference exact module va sign logic.

ortacha sin300° ni exact toping.

💡 Maslahat: Reference angle 60°, QIV sine manfiy.

  1. $\alpha=360^\circ-300^\circ=60^\circ$
  2. QIV’da sine manfiy.

✅ Javob: $\sin300^\circ=-\frac{\sqrt3}{2}$

Nega bu usul ishlaydi: X-axis reflection y ishorasini o‘zgartiradi.

ortacha cos315° ni exact toping.

💡 Maslahat: Reference angle 45°, QIV cosine musbat.

  1. $\alpha=45^\circ$
  2. QIV’da cosine musbat.

✅ Javob: $\cos315^\circ=\frac{\sqrt2}{2}$

Nega bu usul ishlaydi: Reference exact module saqlanadi.

ortacha tan330° ni exact toping.

💡 Maslahat: Reference angle 30°, QIV tangent manfiy.

  1. $\alpha=30^\circ$
  2. $|\tan330^\circ|=\sqrt3/3$
  3. QIV sign negative.

✅ Javob: $\tan330^\circ=-\frac{\sqrt3}{3}$

Nega bu usul ishlaydi: QIV sin−/cos+ ratio manfiy.

murakkab sin(7π/6) ni exact toping.

💡 Maslahat: 7π/6=210°, reference angle π/6.

  1. $7\pi/6=\pi+\pi/6$
  2. QIII’da sine manfiy.
  3. $\sin(\pi/6)=1/2$

✅ Javob: $\sin\frac{7\pi}{6}=-\frac12$

Nega bu usul ishlaydi: Radian special angle reference transferi gradusdagi bilan bir xil.

murakkab cos(5π/4) ni exact toping.

💡 Maslahat: 5π/4=225°, QIII.

  1. Reference angle $\pi/4$.
  2. QIII’da cosine manfiy.

✅ Javob: $\cos\frac{5\pi}{4}=-\frac{\sqrt2}{2}$

Nega bu usul ishlaydi: π/4 exact module + QIII sign.

murakkab tan(5π/3) ni exact toping.

💡 Maslahat: Reference angle π/3, QIV.

  1. $5\pi/3=2\pi-\pi/3$
  2. QIV’da tangent manfiy.
  3. $\tan(\pi/3)=\sqrt3$

✅ Javob: $\tan\frac{5\pi}{3}=-\sqrt3$

Nega bu usul ishlaydi: QIV sign transfer.

murakkab cot(2π/3) ni exact toping.

💡 Maslahat: 2π/3=120°, reference angle π/3.

  1. QII’da cotangent manfiy.
  2. $|\cot(2\pi/3)|=\cot(\pi/3)=\sqrt3/3$

✅ Javob: $\cot\frac{2\pi}{3}=-\frac{\sqrt3}{3}$

Nega bu usul ishlaydi: cot=cos/sin: QII’da negative.

murakkab sec(2π/3) ni exact toping.

💡 Maslahat: sec=1/cos; cos120°=−1/2.

  1. $\cos(2\pi/3)=-1/2$
  2. $\sec(2\pi/3)=1/(-1/2)=-2$

✅ Javob: $\sec\frac{2\pi}{3}=-2$

Nega bu usul ishlaydi: Secant cosine reciprocal.

murakkab csc(7π/6) ni exact toping.

💡 Maslahat: csc=1/sin; sin210°=−1/2.

  1. $\sin(7\pi/6)=-1/2$
  2. $\csc(7\pi/6)=-2$

✅ Javob: $\csc\frac{7\pi}{6}=-2$

Nega bu usul ishlaydi: Cosecant sine reciprocal.

murakkab sin(−π/6) ni exact toping.

💡 Maslahat: −π/6 coterminal 11π/6 yoki sine odd symmetry.

  1. $-\pi/6$ QIV’da, reference angle $\pi/6$.
  2. QIV sine manfiy.

✅ Javob: $\sin(-\frac\pi6)=-\frac12$

Nega bu usul ishlaydi: Negative angle clockwise rotation QIV terminal nuqta beradi.

Muqobil usul: sin(−α)=−sinα.

murakkab cos(−π/3) ni exact toping.

💡 Maslahat: −π/3 coterminal 5π/3; cosine QIV’da musbat.

  1. Reference angle $\pi/3$.
  2. $\cos(\pi/3)=1/2$

✅ Javob: $\cos(-\frac\pi3)=\frac12$

Nega bu usul ishlaydi: Cosine x-coordinate x-axis reflectionda o‘zgarmaydi.

Muqobil usul: cos(−α)=cosα.

murakkab tan(−π/4) ni exact toping.

💡 Maslahat: QIV reference π/4.

  1. $|\tan(-\pi/4)|=1$
  2. QIV tangent manfiy.

✅ Javob: $\tan(-\frac\pi4)=-1$

Nega bu usul ishlaydi: Negative 45° terminal ray IV chorakda.

murakkab sin750° ni exact toping.

💡 Maslahat: 360° karralarini olib tashlang.

  1. $750^\circ-720^\circ=30^\circ$
  2. $\sin750^\circ=\sin30^\circ$

✅ Javob: $\sin750^\circ=\frac12$

Nega bu usul ishlaydi: Coterminal angles bir terminal nuqtaga ega.

murakkab cos(−300°) ni exact toping.

💡 Maslahat: 360° qo‘shing.

  1. $-300^\circ+360^\circ=60^\circ$
  2. $\cos60^\circ=1/2$

✅ Javob: $\cos(-300^\circ)=\frac12$

Nega bu usul ishlaydi: −300° va 60° coterminal.

murakkab tan765° ni exact toping.

💡 Maslahat: 765°−720°=45°.

  1. $765^\circ-2\cdot360^\circ=45^\circ$
  2. $\tan45^\circ=1$

✅ Javob: $\tan765^\circ=1$

Nega bu usul ishlaydi: Full rotations tangent qiymatini saqlaydi.

murakkab 30° uchun sin, cos, tan, cot, sec, csc ning barchasini exact toping.

💡 Maslahat: Avval sin/cos, keyin ratio va reciprocals.

  1. $\sin30^\circ=1/2,\ \cos30^\circ=\sqrt3/2$
  2. $\tan30^\circ=\sqrt3/3,\ \cot30^\circ=\sqrt3$
  3. $\sec30^\circ=2\sqrt3/3,\ \csc30^\circ=2$

✅ Javob: $\left(\frac12,\frac{\sqrt3}{2},\frac{\sqrt3}{3},\sqrt3,\frac{2\sqrt3}{3},2\right)$

Nega bu usul ishlaydi: Barcha six functions bir unit-circle coordinate pair’dan olinadi.

murakkab 135° uchun sin, cos, tan, cot, sec, csc ning barchasini exact toping.

💡 Maslahat: Reference angle 45°, QII signs.

  1. $\sin135^\circ=\sqrt2/2,\ \cos135^\circ=-\sqrt2/2$
  2. $\tan135^\circ=-1,\ \cot135^\circ=-1$
  3. $\sec135^\circ=-\sqrt2,\ \csc135^\circ=\sqrt2$

✅ Javob: $\left(\frac{\sqrt2}{2},-\frac{\sqrt2}{2},-1,-1,-\sqrt2,\sqrt2\right)$

Nega bu usul ishlaydi: 45° exact module va QII signlar reciprocal qiymatlarga ham uzatiladi.

murakkab sin30°+cos60°+tan45° ni exact hisoblang.

💡 Maslahat: Har bir maxsus qiymatni alohida qo‘ying.

  1. $\sin30^\circ=1/2$
  2. $\cos60^\circ=1/2$
  3. $\tan45^\circ=1$
  4. $1/2+1/2+1=2$

✅ Javob: $2$

Nega bu usul ishlaydi: Exact jadval qiymatlari algebraik ifodaga to‘g‘ridan-to‘g‘ri qo‘yiladi.

Umumiy xatolar

❌ sin30° va cos30° qiymatlarini almashtirish.

30° unit-circle nuqta (√3/2,1/2): cosine x, sine y.

✅ P=(cosθ,sinθ) tartibini ishlating.

sin30°=1/2, cos30°=√3/2.

❌ sin60° ni 1/2 deb yozish.

1/2 — cos60° yoki sin30° qiymati.

✅ 30–60–90 triangle’da 60° qarshisidagi uzun katet √3/2.

sin60°=√3/2.

❌ tan30° ni √3 deb olish.

√3 — tan60°.

✅ tan30°=(1/2)/(√3/2)=√3/3.

tan30°=√3/3.

❌ tan90°=∞ deb real qiymat yozish.

tan=sin/cos va cos90°=0; 1/0 aniqlanmagan.

✅ “Aniqlanmagan” deb yozing.

tan90° undefined.

❌ cot0°=0 deb yozish.

cot=cos/sin=1/0.

✅ sinθ=0 bo‘lsa cot va csc aniqlanmagan.

cot0° undefined.

❌ secθ ni inverse cosine deb talqin qilish.

sec reciprocal funksiya; arccos esa inverse function.

✅ secθ=1/cosθ deb ishlating.

sec60°=2, arccos(1/2)=60°.

❌ csc45°=√2/2 deb olish.

Cosecant sine reciprocalidir.

✅ 1/(√2/2)=√2.

csc45°=√2.

❌ II chorakda barcha special qiymatlarni musbat olish.

Reference angle faqat modulni beradi; quadrant sign alohida.

✅ QII: sin+, cos−, tan−.

cos150°=−√3/2.

❌ III chorakda tangentni manfiy qilish.

sin va cos ikkalasi manfiy, nisbat musbat.

✅ tan signni sin/cos ratio’dan oling.

tan225°=1.

❌ IV chorakda cosine’ni manfiy qilish.

IV chorakda x>0.

✅ cosθ=x signini unit circle’dan oling.

cos315°=√2/2.

❌ Reference angle qiymatini original burchak bilan signni tekshirmasdan tenglashtirish.

Simmetriya modulni saqlaydi, signni emas.

✅ Avval exact modul, keyin quadrant sign.

sin330°=−sin30°=−1/2.

❌ 7π/6 ni 7·180/6 gradus deb o‘ylab exact jadval bilan bog‘lamaslik.

Radian special angle gradus ekvivalentiga mos.

✅ 7π/6=210°, reference π/6.

sin7π/6=−1/2.

❌ Katta burchakni jadvalda topolmagach calculatorga o‘tish.

Koterminal reduction exact special angle’ga olib kelishi mumkin.

✅ 360° yoki 2π karralarini olib tashlang.

sin750°=sin30°=1/2.

❌ cos(−θ)=−cosθ deb olish.

Cosine x-axis reflectionda x-coordinate’ni saqlaydi.

✅ Negative angle’ni coterminal/QIV orqali tekshiring.

cos(−60°)=cos60°=1/2.

❌ sin(−θ)=sinθ deb olish.

X-axis reflection y-coordinate signini o‘zgartiradi.

✅ sin(−θ)=−sinθ.

sin(−30°)=−1/2.

❌ 1/√3 ni 1/3 deb “soddalashtirish”.

√3 va 3 bir xil emas.

✅ Rationalize: 1/√3=√3/3.

tan30°=√3/3.

❌ √2/2 ni √(2/2)=1 deb olish.

Radical faqat numerator 2 ustida; kasr strukturasi o‘zgartirilmaydi.

✅ √2/2 exact shaklni saqlang.

sin45°=√2/2≈0.707, 1 emas.

❌ Cofunctionda sin30°=sin60° deb olish.

Complementary juftlarda function nomi almashadi.

✅ sin30°=cos60°, cos30°=sin60°.

Ikkalasi 1/2 yoki √3/2 tarzda mos keladi.

❌ Sec/csc/cot reciprocal qiymatlarda denominator nolni tekshirmaslik.

Reciprocal of zero mavjud emas.

✅ Avval sin/cos/tan denominatorini tekshiring.

sec90°, csc0°, cot0° undefined.

❌ Special exact qiymatni faqat decimal bilan berish.

Decimal rounding keyingi algebraik ishlarda aniqlikni yo‘qotadi.

✅ Exact radical/kasr shaklni asosiy javob qiling; decimal faqat qo‘shimcha.

sin60°=√3/2, taxminan 0.866.

Noto'g'ri tasavvurlar

Maxsus burchak jadvali faqat yodlanadi, chiqarib bo‘lmaydi.

0/90 endpointlari va 30–60–90 hamda 45–45–90 triangles butun first-quadrant jadvalni qayta quradi.

Har quadrant uchun alohida katta jadval yodlash kerak.

Bitta first-quadrant exact jadval + reference angle + quadrant sign yetarli.

Tangent uchun alohida qiymat jadvalini yodlash shart.

tan=sin/cos orqali sine/cosine jadvalidan chiqariladi.

Cotangent, secant va cosecant butunlay yangi mustaqil qiymatlar.

Ular reciprocal/ratio orqali asosiy sin, cos, tan qiymatlardan olinadi.

Undefined trig qiymatni “juda katta son” deb olish mumkin.

0 denominatorli ratio real qiymatga ega emas; undefined va limit tushunchalari alohida.

Reference angle original burchakning ishorasini ham beradi.

Reference angle faqat modul geometriyasini beradi; ishora original quadrantdan keladi.

Radianlarda exact value jadvali boshqa.

π/6,π/4,π/3 aynan 30°,45°,60° burchaklarning boshqa birlikdagi yozuvi.

Manfiy burchak special qiymatga ega bo‘la olmaydi.

Negative angles unit-circle symmetry/coterminal reduction orqali exact baholanadi.

Reciprocal notation sec/csc ni inverse trig functions bilan bir xil qiladi.

sec=1/cos va csc=1/sin; arccos/arcsin esa input-outputni teskari qiluvchi funksiyalar.

Calculator exact special-value geometriyasini almashtiradi.

Calculator verification va nonspecial approximation uchun foydali; exact radicals maxsus triangle/unit-circle’dan keladi.

Amaliy qo'llanilishi

To‘g‘ri burchakli geometriya

30°,45°,60° li uchburchaklarda tomon va balandliklarni calculator ishlatmasdan exact topish.

Koordinata geometriya

Maxsus yo‘nalishdagi unit vector komponentlari cosθ va sinθ exact qiymatlari bilan beriladi.

Mexanika

30°,45°,60° yo‘nalishdagi kuch yoki tezlik komponentlarini exact ajratish.

Kompyuter grafika

Objektlarni 30°,45°,60° ga aylantirishda rotation matrix elementlari exact sin/cos qiymatlardan foydalanadi.

Signal va to‘lqin

Maxsus phase burchaklaridagi sinusoidal signal amplitude komponentlarini exact baholash.

Muhandislik

Standart inclination va slope burchaklarida tan qiymati gradientni exact beradi.

Matematik analizga tayyorgarlik

Trig limit, hosila va integral misollarida special points qiymatlari tez substitution uchun kerak bo‘ladi.

Imtihon va algebraik soddalashtirish

SAT/milliy sertifikat tipidagi trig ifodalarda exact values keyingi identity va equation ishini keskin tezlashtiradi.

Maxsus burchaklar: exact trig qiymatlar xaritasi

MAXSUS BURCHAKLAR — EXACT QIYMATLARθ0°30° / π/645° / π/460° / π/390° / π/2sin01/2√2/2√3/21cos1√3/2√2/21/20tan0√3/31√3undefined30°45°60°Workflow: reference angle → exact modul → quadrant sign → ratio/reciprocal → undefined check

First-quadrant exact sin/cos/tan jadvali ikki special triangle’dan keladi; boshqa quadrantlar reference angle va sign orqali olinadi.

Xulosa

Maxsus burchaklar: 0°,30°,45°,60°,90° ↔ 0,π/6,π/4,π/3,π/2. 30–60–90 nisbat 1:√3:2; 45–45–90 nisbat 1:1:√2. Unit circle’da P=(cosθ,sinθ). Boshqa quadrantlarda reference angle modulni, quadrant ishorani beradi. tan=sin/cos, cot=cos/sin, sec=1/cos, csc=1/sin; denominator 0 bo‘lsa funksiya aniqlanmagan.

Topic66 shu exact qiymatlar va unit-circle geometriyasidan foydalanib Pifagor, quotient va reciprocal trigonometrik ayniyatlarni sistematik quradi. Topic67–71 esa reduction, addition, double-angle va half-angle formulalariga o‘tadi.

Bog'liq mavzular

Oldin bilishingiz kerak: Trigonometriya. Asosiy tushunchalar, To'g'ri burchakli uchburchak, Koordinatalar sistemasi

Bog'liq mavzular: Burchaklar va masofalar, Aylana va doira

Keyingi mavzular: Asosiy trigonometrik ayniyatlar, Keltirish formulalari. Uchburchakda sinuslar va kosinuslar teoremasi, Trigonometrik funksiyalar va ularning xossalari

Manbalar

Ro'yxatdan o'tib, mashq qilishni boshlang