Trigonometriya. Asosiy tushunchalar

Ushbu darsda trigonometriyaning eng dastlabki tushunchalari — burchak, uning o‘lchovi va to‘g‘ri burchakli uchburchakdagi trigonometrik nisbatlar haqida so‘z yuritamiz. Trigonometriya so‘zi yunoncha "trigonon" (uchburchak) va "metron" (o‘lchov) so‘zlaridan olingan, ya’ni bu — uchburchaklarni o‘lchash haqidagi fan.

§1. Burchak va uning o‘lchovi

Burchakni ikki xil birlikda o‘lchaymiz: gradus va radian.

Ta’rif. Aylananing uzunligi radiusiga teng bo‘lgan yoyiga tiralib turgan markaziy burchak $1$ radian deyiladi.

To‘liq aylana $360^\circ$ ga yoki $2\pi$ radianga teng. Demak, asosiy ko‘prik formulasi:

Qoida. $$ 180^\circ = \pi \text{ radian}, \qquad 1^\circ = \frac{\pi}{180} \text{ rad}, \qquad 1 \text{ rad} = \frac{180^\circ}{\pi}. $$

Savol. Nega aynan radian qulay? Chunki radianda yoy uzunligi oddiygina $l = r\varphi$ ko‘rinishida yoziladi — hech qanday $\pi/180$ ko‘paytuvchisiz.

Misol. $60^\circ$ ni radianga aylantiring.
Yechish. Har bir gradusni $\frac{\pi}{180}$ ga ko‘paytiramiz:

$$60^\circ = 60 \cdot \frac{\pi}{180} = \frac{60\pi}{180} = \frac{\pi}{3}.$$

Javob. $\dfrac{\pi}{3}$.

§2. To‘g‘ri burchakli uchburchakda nisbatlar

To‘g‘ri burchakli uchburchakni olamiz. $C$ uchidagi burchak $90^\circ$, biz esa o‘tkir $A$ burchakni ($\alpha$) kuzatamiz.

$\alpha$ burchak uchun tomonlar shunday nomlanadi: qarshi katet $a$ (burchak qarshisida), yaqin katet $b$ va gipotenuza $c$ (to‘g‘ri burchak qarshisida, eng uzun tomon).

Ta’rif. O‘tkir $\alpha$ burchakning trigonometrik nisbatlari:

$$\sin\alpha = \frac{a}{c}=\frac{\text{qarshi}}{\text{gipotenuza}}, \quad \cos\alpha = \frac{b}{c}=\frac{\text{yaqin}}{\text{gipotenuza}},$$

$$\tan\alpha = \frac{a}{b}=\frac{\sin\alpha}{\cos\alpha}, \quad \cot\alpha=\frac{b}{a}=\frac{\cos\alpha}{\sin\alpha}.$$

Misol. Katetlari $a=3$, $b=4$ bo‘lgan to‘g‘ri burchakli uchburchakda $\sin\alpha$, $\cos\alpha$, $\tan\alpha$ ni toping.
Yechish. Avval gipotenuzani Pifagor teoremasidan topamiz:

$$c=\sqrt{a^2+b^2}=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5.$$

Endi ta’riflarga qo‘yamiz:

$$\sin\alpha=\frac{3}{5}=0{,}6,\quad \cos\alpha=\frac{4}{5}=0{,}8,\quad \tan\alpha=\frac{3}{4}=0{,}75.$$

Javob. $\sin\alpha=0{,}6$, $\cos\alpha=0{,}8$, $\tan\alpha=0{,}75$.

§3. Birlik aylanada burchak

Katta (o‘tmas, manfiy, $360^\circ$ dan katta) burchaklar uchun ta’rifni birlik aylana yordamida kengaytiramiz — radiusi $1$ ga teng, markazi koordinata boshida joylashgan aylana.

Ta’rif. $Ox$ o‘qining musbat yo‘nalishidan $\alpha$ burchak burilganda aylana bilan kesishgan nuqta $P(x;y)$ bo‘lsin. Shunda $\cos\alpha=x$, $\sin\alpha=y$.

Eslatma. Birlik aylanada radius $1$ bo‘lgani uchun $\sin\alpha$ va $\cos\alpha$ hamisha $-1$ bilan $1$ oralig‘ida bo‘ladi: $-1\le\sin\alpha\le1$. Bu chegaradan chiqib ketgan javob — xatolik belgisi.

Misol. $P$ nuqta $x=y$ bo‘lgan, birinchi chorakdagi aylana nuqtasi bo‘lsa, $\alpha$ ni toping.
Yechish. $x^2+y^2=1$ va $x=y$ dan $2x^2=1$, $x=\frac{1}{\sqrt2}$. Bu $\cos\alpha=\sin\alpha$ degani, ya’ni $\alpha=45^\circ$.
Javob. $\alpha=45^\circ=\dfrac{\pi}{4}$.

§ Lug‘at

burchak — angle
radian — radian
gradus — degree
gipotenuza — hypotenuse
katet — leg (of a right triangle)
sinus — sine
kosinus — cosine
birlik aylana — unit circle