Boshlang'ich tushunchalar: radian, birlik aylana, ishoralar: matematika darsi

$\sin 2$ va $\sin 2^\circ$ ikki xil son: birinchisi taxminan $0{,}909$ , ikkinchisi taxminan $0{,}035$ , chunki gradus belgisi bo'lmasa burchak radianda o'lchanadi. Bu darsda burchak o'lchovlari, trigonometrik funksiyalarning birlik aylanadagi ta'rifi, ular qachon mavjudligi, jadval qiymatlari va choraklar bo'yicha ishoralar to'liq beriladi. Keyingi hamma darslar shularga tayanadi.

Burish burchagi

Trigonometriyada burchak nurni koordinata boshi atrofida burish bilan hosil bo'ladi. Nur $Ox$ o'qining musbat yo'nalishidan boshlab buriladi. Soat strelkasiga teskari burish musbat burchak, soat strelkasi bo'yicha burish manfiy burchak beradi. Burchak istalgan kattalikda bo'lishi mumkin: $390^\circ$ bu bir to'liq aylana va yana $30^\circ$ , shuning uchun $390^\circ$ va $30^\circ$ nurni bir xil holatga olib keladi. $-30^\circ$ va $330^\circ$ ham bir xil holat beradi.

$\alpha$ va $\alpha+360^\circ\cdot n$ ( $n\in Z$ ) burchaklar nurni bir xil holatga keltiradi, shuning uchun ularning barcha trigonometrik funksiyalari teng.

Radian

Bir radian deb uzunligi radiusga teng yoy tiraladigan markaziy burchakka aytiladi. Butun aylana uzunligi $2\pi R$ , ya'ni $2\pi$ ta radius, demak to'liq aylana $2\pi$ radian.

180^\circ=\pi\ \text{rad}

Gradus va radian orasidagi asosiy moslik

\alpha_{\text{rad}}=\alpha^\circ\cdot\frac{\pi}{180}

\alpha^\circ=\alpha_{\text{rad}}\cdot\frac{180}{\pi}

Graduslardan radianga va radiandan graduslarga o'tish

$\pi$ soni $\approx 3{,}14$ bo'lgani uchun $1$ radian $\approx 57{,}3^\circ$ va $1^\circ\approx 0{,}0175$ radian. Masalan:

240^\circ=240\cdot\frac{\pi}{180}=\frac{4\pi}{3}

\frac{7\pi}{9}=\frac{7\cdot 180^\circ}{9}=140^\circ

Gradus	$30^\circ$	$45^\circ$	$60^\circ$	$90^\circ$
GradusRadian	$30^\circ$ $\dfrac{\pi}{6}$	$45^\circ$ $\dfrac{\pi}{4}$	$60^\circ$ $\dfrac{\pi}{3}$	$90^\circ$ $\dfrac{\pi}{2}$

Gradus	$120^\circ$	$135^\circ$	$150^\circ$	$180^\circ$
GradusRadian	$120^\circ$ $\dfrac{2\pi}{3}$	$135^\circ$ $\dfrac{3\pi}{4}$	$150^\circ$ $\dfrac{5\pi}{6}$	$180^\circ$ $\pi$

Gradus	$210^\circ$	$225^\circ$	$240^\circ$	$270^\circ$
GradusRadian	$210^\circ$ $\dfrac{7\pi}{6}$	$225^\circ$ $\dfrac{5\pi}{4}$	$240^\circ$ $\dfrac{4\pi}{3}$	$270^\circ$ $\dfrac{3\pi}{2}$

Gradus	$300^\circ$	$315^\circ$	$330^\circ$	$360^\circ$
GradusRadian	$300^\circ$ $\dfrac{5\pi}{3}$	$315^\circ$ $\dfrac{7\pi}{4}$	$330^\circ$ $\dfrac{11\pi}{6}$	$360^\circ$ $2\pi$

Birlik aylanadagi asosiy burchaklar. Har bir nuqta o'z o'rnida:

30^\circ

nuqtaning koordinatalari

\left(\dfrac{\sqrt{3}}{2};\,\dfrac{1}{2}\right)

Yoy uzunligi va sektor yuzi

l=R\alpha

S=\frac{1}{2}R^2\alpha=\frac{1}{2}lR

Faqat

\alpha

radianda bo'lganda

$R=6$ , $\alpha=\dfrac{2\pi}{3}$ bo'lsa:

l=6\cdot\frac{2\pi}{3}=4\pi

S=\frac{1}{2}\cdot 36\cdot\frac{2\pi}{3}=12\pi

Yoy uzunligi va sektor yuzi

Birlik aylana va funksiyalarning ta'rifi

Birlik aylana deb markazi koordinata boshida, radiusi $1$ ga teng aylanaga aytiladi. $A(1;\,0)$ nuqtani $\alpha$ burchakka bursak, aylanada $M(x;\,y)$ nuqta hosil bo'ladi.

M

nuqtaning abssissasi

\cos\alpha

, ordinatasi

\sin\alpha

Funksiya	Ta'rifi	Qachon mavjud
Funksiya $\sin\alpha$	Ta'rifi $y$ , nuqtaning ordinatasi	Qachon mavjudhar qanday $\alpha$
Funksiya $\cos\alpha$	Ta'rifi $x$ , nuqtaning abssissasi	Qachon mavjudhar qanday $\alpha$
Funksiya $\operatorname{tg}\alpha$	Ta'rifi $\dfrac{y}{x}=\dfrac{\sin\alpha}{\cos\alpha}$	Qachon mavjud $\cos\alpha\ne 0$ , ya'ni $\alpha\ne\dfrac{\pi}{2}+\pi n$
Funksiya $\operatorname{ctg}\alpha$	Ta'rifi $\dfrac{x}{y}=\dfrac{\cos\alpha}{\sin\alpha}$	Qachon mavjud $\sin\alpha\ne 0$ , ya'ni $\alpha\ne\pi n$

O'tkir burchak uchun bu ta'rif to'g'ri burchakli uchburchakdagi ta'rif bilan bir xil: sinus qarshi katetning gipotenuzaga, kosinus yopishgan katetning gipotenuzaga, tangens qarshi katetning yopishgan katetga, kotangens yopishgan katetning qarshi katetga nisbati.

$\operatorname{tg}90^\circ$ , $\operatorname{tg}270^\circ$ , $\operatorname{ctg}0^\circ$ , $\operatorname{ctg}180^\circ$ , $\operatorname{ctg}360^\circ$ mavjud emas: maxraj nolga teng. Ular $0$ ham, cheksiz son ham emas.

Qiymatlar chegarasi

Nuqta radiusi $1$ bo'lgan aylanada yotgani uchun uning koordinatalari $-1$ va $1$ orasida bo'ladi.

-1\le\sin\alpha\le 1

-1\le\cos\alpha\le 1

\operatorname{tg}\alpha

\operatorname{ctg}\alpha

esa istalgan haqiqiy son bo'la oladi

Shuning uchun $\sin\alpha=1{,}2$ tenglik hech qanday burchakda bajarilmaydi, $\operatorname{tg}\alpha=1{,}2$ esa bajariladi. Bu chegara boshqa baholarni ham beradi: $|\sin\alpha|\le 1<\dfrac{\pi}{2}$ , ya'ni $\sin\alpha$ soni radian o'lchovida $\left(-\dfrac{\pi}{2};\,\dfrac{\pi}{2}\right)$ oraliqda yotadi, shuning uchun har qanday $\alpha$ da $\cos(\sin\alpha)>0$ . $\sin(\sin\alpha)$ uchun esa bunday xulosa yo'q: uning ishorasi $\sin\alpha$ ning ishorasi bilan bir xil.

Aylana bo'ylab burish

Radiusi $R$ bo'lgan aylanada $\alpha$ burchakka mos nuqta $(R\cos\alpha;\,R\sin\alpha)$ dir. Koordinata boshi atrofida burishda nuqta shu qoida bilan ko'chadi.

Burish	$(x;\,y)$ nuqta o'tadi	$(3;\,4)$ uchun
Burish $90^\circ$	$(x;\,y)$ nuqta o'tadi $(-y;\,x)$	$(3;\,4)$ uchun $(-4;\,3)$
Burish $180^\circ$	$(x;\,y)$ nuqta o'tadi $(-x;\,-y)$	$(3;\,4)$ uchun $(-3;\,-4)$
Burish $270^\circ$ yoki $-90^\circ$	$(x;\,y)$ nuqta o'tadi $(y;\,-x)$	$(3;\,4)$ uchun $(4;\,-3)$
Burish $\varphi$	$(x;\,y)$ nuqta o'tadi $(x\cos\varphi-y\sin\varphi;\,x\sin\varphi+y\cos\varphi)$	$(3;\,4)$ uchun $\varphi=90^\circ$ da birinchi qator

Jadval qiymatlari

$30^\circ$ va $60^\circ$ qiymatlari tomonlari $1,\ \sqrt{3},\ 2$ bo'lgan uchburchakdan, $45^\circ$ qiymatlari tomonlari $1,\ 1,\ \sqrt{2}$ bo'lgan uchburchakdan chiqadi. Sinusni eslab qolish oson: $0^\circ,\ 30^\circ,\ 45^\circ,\ 60^\circ,\ 90^\circ$ da u $\dfrac{\sqrt{0}}{2},\ \dfrac{\sqrt{1}}{2},\ \dfrac{\sqrt{2}}{2},\ \dfrac{\sqrt{3}}{2},\ \dfrac{\sqrt{4}}{2}$ ga teng. Kosinusda shu qator teskari o'qiladi. Jadvalda "yo'q" so'zi qiymat mavjud emasligini bildiradi.

$\alpha$	$0^\circ$	$30^\circ$	$45^\circ$	$60^\circ$	$90^\circ$
$\alpha$ $\sin\alpha$	$0^\circ$ $0$	$30^\circ$ $\dfrac{1}{2}$	$45^\circ$ $\dfrac{\sqrt{2}}{2}$	$60^\circ$ $\dfrac{\sqrt{3}}{2}$	$90^\circ$ $1$
$\alpha$ $\cos\alpha$	$0^\circ$ $1$	$30^\circ$ $\dfrac{\sqrt{3}}{2}$	$45^\circ$ $\dfrac{\sqrt{2}}{2}$	$60^\circ$ $\dfrac{1}{2}$	$90^\circ$ $0$
$\alpha$ $\operatorname{tg}\alpha$	$0^\circ$ $0$	$30^\circ$ $\dfrac{\sqrt{3}}{3}$	$45^\circ$ $1$	$60^\circ$ $\sqrt{3}$	$90^\circ$ yo'q
$\alpha$ $\operatorname{ctg}\alpha$	$0^\circ$ yo'q	$30^\circ$ $\sqrt{3}$	$45^\circ$ $1$	$60^\circ$ $\dfrac{\sqrt{3}}{3}$	$90^\circ$ $0$

Koordinata o'qlaridagi burchaklar uchun qiymatni aylanadagi nuqtadan to'g'ridan-to'g'ri o'qiymiz: $0^\circ$ da $(1;\,0)$ , $90^\circ$ da $(0;\,1)$ , $180^\circ$ da $(-1;\,0)$ , $270^\circ$ da $(0;\,-1)$ .

$\alpha$	$0^\circ$	$90^\circ$	$180^\circ$	$270^\circ$	$360^\circ$
$\alpha$ $\sin\alpha$	$0^\circ$ $0$	$90^\circ$ $1$	$180^\circ$ $0$	$270^\circ$ $-1$	$360^\circ$ $0$
$\alpha$ $\cos\alpha$	$0^\circ$ $1$	$90^\circ$ $0$	$180^\circ$ $-1$	$270^\circ$ $0$	$360^\circ$ $1$
$\alpha$ $\operatorname{tg}\alpha$	$0^\circ$ $0$	$90^\circ$ yo'q	$180^\circ$ $0$	$270^\circ$ yo'q	$360^\circ$ $0$
$\alpha$ $\operatorname{ctg}\alpha$	$0^\circ$ yo'q	$90^\circ$ $0$	$180^\circ$ yo'q	$270^\circ$ $0$	$360^\circ$ yo'q

$0^\circ$ dan $360^\circ$ gacha bo'lgan barcha jadval burchaklarining to'liq ro'yxati keltirish formulalari darsida beriladi. Maxrajda ildiz bilan yozilgan qiymat o'sha son: $\dfrac{1}{\sqrt{3}}=\dfrac{\sqrt{3}}{3}$ , $\dfrac{1}{\sqrt{2}}=\dfrac{\sqrt{2}}{2}$ . Jadval qiymatlarining bir qismi irratsional, bir qismi ratsional:

\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}

\sin\frac{\pi}{6}=\frac{1}{2}

Birinchisi irratsional, ikkinchisi ratsional son

Choraklar bo'yicha ishoralar

Sinus ishorasi $y$ ning, kosinus ishorasi $x$ ning ishorasi bilan bir xil. Tangens va kotangens $x$ va $y$ ning nisbati, shuning uchun ular $x,\ y$ bir xil ishorali bo'lganda musbat.

Chorak	Burchak
I	$0<\alpha<\dfrac{\pi}{2}$
II	$\dfrac{\pi}{2}<\alpha<\pi$
III	$\pi<\alpha<\dfrac{3\pi}{2}$
IV	$\dfrac{3\pi}{2}<\alpha<2\pi$

Chorak	$\sin$	$\cos$	$\operatorname{tg}$	$\operatorname{ctg}$
ChorakI	$\sin$ $+$	$\cos$ $+$	$\operatorname{tg}$ $+$	$\operatorname{ctg}$ $+$
ChorakII	$\sin$ $+$	$\cos$ $-$	$\operatorname{tg}$ $-$	$\operatorname{ctg}$ $-$
ChorakIII	$\sin$ $-$	$\cos$ $-$	$\operatorname{tg}$ $+$	$\operatorname{ctg}$ $+$
ChorakIV	$\sin$ $-$	$\cos$ $+$	$\operatorname{tg}$ $-$	$\operatorname{ctg}$ $-$

Katta burchakda avval to'liq aylanalarni tashlab, chorakni aniqlaymiz: $530^\circ=360^\circ+170^\circ$ II chorakda, $-100^\circ=-360^\circ+260^\circ$ III chorakda. O'qlardagi burchaklar hech qaysi chorakka kirmaydi: ularda funksiya $0$ , $\pm 1$ yoki mavjud emas.

Radianda berilgan sonlarning choragi

Gradus belgisi bo'lmasa, son radian. Chorak chegaralarini taqqoslaymiz: $\dfrac{\pi}{2}\approx 1{,}57$ , $\pi\approx 3{,}14$ , $\dfrac{3\pi}{2}\approx 4{,}71$ , $2\pi\approx 6{,}28$ .

Son	$1$	$2$	$3$	$4$	$5$	$6$
SonChorak	$1$ I	$2$ II	$3$ II	$4$ III	$5$ IV	$6$ IV
Son $\sin$	$1$ $+$	$2$ $+$	$3$ $+$	$4$ $-$	$5$ $-$	$6$ $-$
Son $\cos$	$1$ $+$	$2$ $-$	$3$ $-$	$4$ $-$	$5$ $+$	$6$ $+$
Son $\operatorname{tg}$	$1$ $+$	$2$ $-$	$3$ $-$	$4$ $+$	$5$ $-$	$6$ $-$

$6{,}28$ dan katta sonda avval $2\pi$ ni ayiramiz: $7-2\pi\approx 0{,}72$ , demak $7$ radian I chorakda. Manfiy son soat strelkasi bo'yicha sanaladi: $-1$ radian IV chorakda, $-2$ radian III chorakda.

Juft-toqlik va davriylik

$\alpha$ va $-\alpha$ burchaklarga mos nuqtalar $Ox$ o'qiga nisbatan simmetrik: $x$ bir xil, $y$ qarama-qarshi.

\sin(-\alpha)=-\sin\alpha

\cos(-\alpha)=\cos\alpha

Kosinus juft, sinus toq

\operatorname{tg}(-\alpha)=-\operatorname{tg}\alpha

\operatorname{ctg}(-\alpha)=-\operatorname{ctg}\alpha

Tangens va kotangens toq

\sin(\alpha+2\pi n)=\sin\alpha

\cos(\alpha+2\pi n)=\cos\alpha

n\in Z

; sinus va kosinusning eng kichik musbat davri

2\pi

\operatorname{tg}(\alpha+\pi n)=\operatorname{tg}\alpha

\operatorname{ctg}(\alpha+\pi n)=\operatorname{ctg}\alpha

n\in Z

; tangens va kotangensning eng kichik musbat davri

\pi

Ifoda qachon ma'noga ega

Trigonometrik son ildiz yoki logarifm ostida kelsa, uning ishorasini chorakdan aniqlaymiz. Ildiz ostidagi son manfiy bo'lmasligi, logarifm ostidagi son musbat bo'lishi, maxraj nolga teng bo'lmasligi kerak.

Logarifmning ishorasini ham shu yerda bilish kerak bo'ladi: $a>1$ da $x>1$ bo'lsa $\log_a x>0$ , $0<x<1$ bo'lsa $\log_a x<0$ ; $0<a<1$ da aksincha. Masalan, $\log_2 1{,}43>0$ , $\log_{1/2}0{,}7>0$ , $\ln 0{,}5<0$ .

Sonlarni taqqoslashda oraliq qiymatlardan foydalanamiz. Uchta sonni olamiz:

$a=\cos\dfrac{2\pi}{3}=-\dfrac{1}{2}<0$ .
$b=\log_3\dfrac{4\pi}{3}$ : $\dfrac{4\pi}{3}\approx 4{,}19$ soni $3$ va $9$ orasida, demak $1<b<2$ va aniqrog'i $b\approx 1{,}30$ .
$c=2^{\pi/3}$ : $\dfrac{\pi}{3}>1$ , demak $c>2$ .

Shuning uchun $a<b<c$ .

Ishlangan misollar

1-misol

Hisoblang:

6\sin 30^\circ-4\cos 60^\circ

+\sqrt{3}\operatorname{tg}60^\circ

$6\sin 30^\circ=6\cdot\frac{1}{2}=3$
Jadvaldan: $\sin 30^\circ=\dfrac{1}{2}$ .
$4\cos 60^\circ=4\cdot\frac{1}{2}=2$
Jadvaldan: $\cos 60^\circ=\dfrac{1}{2}$ .
$\sqrt{3}\operatorname{tg}60^\circ=\sqrt{3}\cdot\sqrt{3}=3$
Jadvaldan: $\operatorname{tg}60^\circ=\sqrt{3}$ .
$3-2+3=4$
Natijalarni qo'shamiz.

Javob: $4$

2-misol

Hisoblang:

3\sin 90^\circ-4\cos 180^\circ

+2\operatorname{tg}0^\circ-7\cos 270^\circ

$\sin 90^\circ=1,\quad \cos 180^\circ=-1$
$90^\circ$ da nuqta $(0;\,1)$ , $180^\circ$ da $(-1;\,0)$ .
$\operatorname{tg}0^\circ=0,\quad \cos 270^\circ=0$
$0^\circ$ da $y=0$ , $270^\circ$ da $x=0$ .
$3\cdot 1-4\cdot(-1)+0-0=7$
Minusga minus plus beradi.

Javob: $7$

3-misol

$\sin 3\cdot\cos 5\cdot\operatorname{tg}2$ ko'paytmaning ishorasini aniqlang (burchaklar radianda).

$1{,}57<3<3{,}14:\quad \sin 3>0$
$3$ radian II chorakda, u yerda sinus musbat.
$4{,}71<5<6{,}28:\quad \cos 5>0$
$5$ radian IV chorakda, u yerda kosinus musbat.
$1{,}57<2<3{,}14:\quad \operatorname{tg}2<0$
$2$ radian II chorakda, u yerda tangens manfiy.
$(+)\cdot(+)\cdot(-)=(-)$
Bitta manfiy ko'paytuvchi bor.

Javob: manfiy

4-misol

$a=\sin 460^\circ$ , $b=\cos 590^\circ$ , $c=\operatorname{tg}850^\circ$ , $d=\operatorname{ctg}(-300^\circ)$ sonlaridan qaysilari musbat?

$460^\circ=360^\circ+100^\circ:\quad a>0$
II chorakda sinus musbat.
$590^\circ=360^\circ+230^\circ:\quad b<0$
III chorakda kosinus manfiy.
$850^\circ=720^\circ+130^\circ:\quad c<0$
II chorakda tangens manfiy.
$-300^\circ=-360^\circ+60^\circ:\quad d>0$
I chorakda kotangens musbat.

Javob: $a$ va $d$

5-misol

Qaysi ifoda ma'noga ega? 1) $\log_2\cos\dfrac{7\pi}{5}$ ; 2) $\sqrt{\sin\dfrac{11\pi}{12}}$ ; 3) $\lg\operatorname{ctg}\dfrac{5\pi}{6}$ .

$\frac{7\pi}{5}=252^\circ:\quad \cos\frac{7\pi}{5}<0$
III chorak. Logarifm ostida manfiy son, birinchi ifoda ma'noga ega emas.
$\frac{11\pi}{12}=165^\circ:\quad \sin\frac{11\pi}{12}>0$
II chorak. Ildiz ostida musbat son, ikkinchi ifoda ma'noga ega.
$\frac{5\pi}{6}=150^\circ:\quad \operatorname{ctg}\frac{5\pi}{6}<0$
II chorak. Logarifm ostida manfiy son, uchinchi ifoda ma'noga ega emas.

Javob: faqat 2)

6-misol

Hisoblang:

(9^2)^3:(3^5)^2

+8\sqrt{3}\cdot\operatorname{tg}\frac{\pi}{6}\cdot\cos\frac{\pi}{3}

+\log_2 5\cdot\log_5 32

$(9^2)^3:(3^5)^2=3^{12}:3^{10}=9$
Hammasini $3$ asosiga keltiramiz: $9^6=3^{12}$ .
$8\sqrt{3}\cdot\frac{\sqrt{3}}{3}\cdot\frac{1}{2}=4$
$\operatorname{tg}\dfrac{\pi}{6}=\dfrac{\sqrt{3}}{3}$ , $\cos\dfrac{\pi}{3}=\dfrac{1}{2}$ .
$\log_2 5\cdot\log_5 32=\log_2 32=5$
$\log_a b\cdot\log_b c=\log_a c$ .
$9+4+5=18$
Uchala natijani qo'shamiz.

Javob: $18$

Ko'p uchraydigan xatolar

Xato	To'g'risi
$\sin 2$ ni $\sin 2^\circ$ deb hisoblash	$2$ radian, II chorak, $\sin 2\approx 0{,}909$
$\operatorname{tg}90^\circ=0$	$\operatorname{tg}90^\circ$ mavjud emas, $\operatorname{ctg}90^\circ=0$
$\cos 180^\circ=0$	$\cos 180^\circ=-1$ , $\sin 180^\circ=0$
III chorakda $\operatorname{tg}\alpha<0$	III chorakda $\sin$ va $\cos$ manfiy, nisbati $\operatorname{tg}\alpha>0$
$\cos(-\alpha)=-\cos\alpha$	$\cos(-\alpha)=\cos\alpha$ , kosinus juft
$150^\circ=\dfrac{5\pi}{3}$	$150^\circ=\dfrac{150\pi}{180}=\dfrac{5\pi}{6}$
$\sin\alpha=1{,}2$ dan burchak qidirish	$\|\sin\alpha\|\le 1$ , bunday burchak yo'q
$l=R\alpha$ ga $\alpha$ ni gradusda qo'yish	formulada $\alpha$ faqat radianda

Eslab qoling

$180^\circ=\pi$ radian; $l=R\alpha$ , $S=\dfrac{1}{2}R^2\alpha$ , $\alpha$ radianda.
$\cos\alpha=x$ , $\sin\alpha=y$ ; ikkalasi $[-1;\,1]$ da.
$\operatorname{tg}\alpha$ uchun $\cos\alpha\ne 0$ , $\operatorname{ctg}\alpha$ uchun $\sin\alpha\ne 0$ .
Ishoralar: I hammasi $+$ ; II faqat $\sin$ ; III faqat $\operatorname{tg},\ \operatorname{ctg}$ ; IV faqat $\cos$ .
$\pi\approx 3{,}14$ , $\dfrac{\pi}{2}\approx 1{,}57$ , $\dfrac{3\pi}{2}\approx 4{,}71$ , $2\pi\approx 6{,}28$ .
$\cos$ juft; $\sin,\ \operatorname{tg},\ \operatorname{ctg}$ toq. Davrlar: $\sin,\ \cos$ uchun $2\pi$ ; $\operatorname{tg},\ \operatorname{ctg}$ uchun $\pi$ .