Trigonometrik funksiya qiymatlari

Ushbu darsda trigonometrik funksiyalarning muhim burchaklardagi qiymatlari, ularning ishorasi choraklarga qarab qanday o‘zgarishi va qiymatlar jadvalini yod olishning oson yo‘li haqida so‘z yuritamiz.

§1. Maxsus burchaklar jadvali

Amaliyotda eng ko‘p uchraydigan burchaklar — $0^\circ,\,30^\circ,\,45^\circ,\,60^\circ,\,90^\circ$. Ularning qiymatlarini yod olish shart.

Qoida. Asosiy qiymatlar jadvali:

$$\begin{array}{c|ccccc} \alpha & 0^\circ & 30^\circ & 45^\circ & 60^\circ & 90^\circ\\ \hline \sin\alpha & 0 & \frac{1}{2} & \frac{\sqrt2}{2} & \frac{\sqrt3}{2} & 1\\[4pt] \cos\alpha & 1 & \frac{\sqrt3}{2} & \frac{\sqrt2}{2} & \frac{1}{2} & 0\\[4pt] \tan\alpha & 0 & \frac{\sqrt3}{3} & 1 & \sqrt3 & - \end{array}$$

Eslatma. Jadvalni yod olishning oson usuli: $\sin$ qatoriga $\sqrt0,\sqrt1,\sqrt2,\sqrt3,\sqrt4$ ni yozib, hammasini $2$ ga bo‘ling: $\frac{\sqrt0}{2}=0,\ \frac{\sqrt1}{2}=\frac12,\ \frac{\sqrt2}{2},\ \frac{\sqrt3}{2},\ \frac{\sqrt4}{2}=1$. $\cos$ qatori esa aynan teskari tartibda.

Misol. $\sin30^\circ+\cos60^\circ$ ni hisoblang.
Yechish. Jadvaldan qiymatlarni olamiz:

$$\sin30^\circ+\cos60^\circ=\frac{1}{2}+\frac{1}{2}=1.$$

Javob. $1$.

§2. Choraklarda ishoralar

Ta’rif. Birlik aylanani koordinata o‘qlari to‘rt qismga ajratadi; bu qismlar soat mili teskarisida I, II, III, IV choraklar deb ataladi.

Birlik aylana koordinata o‘qlari bilan to‘rt chorakka bo‘linadi. Funksiya ishorasi nuqtaning $x$ va $y$ koordinatalari ishorasiga bog‘liq, chunki $\cos\alpha=x$, $\sin\alpha=y$.

Qoida. I chorakda barcha funksiyalar musbat; II chorakda faqat $\sin$ musbat; III chorakda faqat $\tan$ (va $\cot$) musbat; IV chorakda faqat $\cos$ musbat. Yodlash iborasi: "Barcha Sinovni Tark etib Ketdi".

Savol. $\cos120^\circ$ musbatmi yoki manfiy? $120^\circ$ II chorakda, u yerda $\cos$ manfiy — demak, javob manfiy bo‘lishi kerak.

Misol. $\cos150^\circ$ ni toping.
Yechish. $150^\circ$ II chorakda, unga tegishli o‘tkir burchak $180^\circ-150^\circ=30^\circ$. II chorakda kosinus manfiy:

$$\cos150^\circ=-\cos30^\circ=-\frac{\sqrt3}{2}.$$

Javob. $-\dfrac{\sqrt3}{2}$.

§3. Katta burchaklar bilan ishlash

$360^\circ$ dan katta yoki manfiy burchaklarda funksiya qiymati $360^\circ$ (yoki $2\pi$) davriylik xossasiga ko‘ra takrorlanadi.

Qoida. $$ \sin(\alpha+360^\circ)=\sin\alpha,\qquad \cos(\alpha+360^\circ)=\cos\alpha. $$

Misol. $\sin405^\circ$ ni hisoblang.
Yechish. $405^\circ$ dan to‘liq aylanani ayiramiz:

$$\sin405^\circ=\sin(405^\circ-360^\circ)=\sin45^\circ=\frac{\sqrt2}{2}.$$

Javob. $\dfrac{\sqrt2}{2}$.

Misol. $\tan60^\circ-\sqrt3\cdot\cos^2 30^\circ$ ni hisoblang.
Yechish. Jadvaldan: $\tan60^\circ=\sqrt3$, $\cos30^\circ=\frac{\sqrt3}{2}$, demak $\cos^2 30^\circ=\frac34$.

$$\sqrt3-\sqrt3\cdot\frac{3}{4}=\sqrt3\left(1-\frac34\right)=\sqrt3\cdot\frac14=\frac{\sqrt3}{4}.$$

Javob. $\dfrac{\sqrt3}{4}$.

§ Lug‘at

qiymat — value
chorak — quadrant
ishora — sign
davriylik — periodicity
jadval — table
o‘tkir burchak — acute angle
o‘tmas burchak — obtuse angle