Trigonometrik tengsizliklar

Bu nima va nega kerak

Fizikada tebranishlar va o'zgaruvchan tok parametrlarini o'rganishda jarayonning ma'lum oraliqda bo'lish vaqtini topish talab etiladi.

Masalan, tarmoqdagi kuchlanish $u(t) = 220\sqrt{2}\sin(100\pi t)$ bo'lsa, uning qachon $110\sqrt{2}\text{ V}$ dan yuqori bo'lishini aniqlash uchun $\sin(100\pi t) > 0{,}5$ tengsizlikni yechish kerak.

Noma'lum burchak trigonometrik funksiya belgisi ostida qatnashgan tengsizliklar trigonometrik tengsizliklar deyiladi. Ularni yechishda eng qulay usul — birlik aylanadan foydalanishdir.

Asosiy g'oya

Birlik aylanada $x = \cos\alpha$ (abssissa) va $y = \sin\alpha$ (ordinata). Tengsizlikni yechish aylananing mos yoyini aniqlash demakdir ($k \in \mathbb{Z}$).

1. Sinusli sodda tengsizliklar ($|a| < 1$)

$y = a$ gorizontal to'g'ri chizig'i aylanani kesadi:

• $\sin x > a$ (yuqori yoy): $\arcsin a + 2\pi k < x < \pi - \arcsin a + 2\pi k.$
• $\sin x < a$ (pastki yoy): $-\pi - \arcsin a + 2\pi k < x < \arcsin a + 2\pi k.$

2. Kosinusli sodda tengsizliklar ($|a| < 1$)

$x = a$ vertikal to'g'ri chizig'i aylanani kesadi:

• $\cos x > a$ (o'ng yoy): $-\arccos a + 2\pi k < x < \arccos a + 2\pi k.$
• $\cos x < a$ (chap yoy): $\arccos a + 2\pi k < x < 2\pi - \arccos a + 2\pi k.$

3. Tangens va kotangensli tengsizliklar

Tangens va kotangensning davri $\pi$ bo'lib, ular bitta asosiy davrda yechiladi:

• $\operatorname{tg} x > a$: $\operatorname{arctg} a + \pi k < x < \frac{\pi}{2} + \pi k$.
• $\operatorname{tg} x < a$: $-\frac{\pi}{2} + \pi k < x < \operatorname{arctg} a + \pi k$.
• $\operatorname{ctg} x > a$: $\pi k < x < \operatorname{arcctg} a + \pi k$.
• $\operatorname{ctg} x < a$: $\operatorname{arcctg} a + \pi k < x < \pi + \pi k$.

Yechilgan misollar

1-misol. Sinusli noqat'iy tengsizlik

$\sin x \ge \frac{1}{2}$

tengsizlikni yeching.

• Birlik aylanada $y = \dfrac{1}{2}$ to'g'ri chiziq o'tkazamiz. Kesishish nuqtalari: $x_1 = \dfrac{\pi}{6}$ va $x_2 = \dfrac{5\pi}{6}$.
• $\sin x \ge \dfrac{1}{2}$ sharti ushbu chiziqdan yuqoridagi yoyga mos keladi.
• Asosiy davrdagi yechim: $\left[\dfrac{\pi}{6}; \dfrac{5\pi}{6}\right]$.
• Davr $2\pi k$ ni qo'shamiz: $\frac{\pi}{6} + 2\pi k \le x \le \frac{5\pi}{6} + 2\pi k, \quad k \in \mathbb{Z}.$

Javob: $\left[\dfrac{\pi}{6} + 2\pi k; \; \dfrac{5\pi}{6} + 2\pi k\right], \quad k \in \mathbb{Z}$.

2-misol. Kosinusli qat'iy tengsizlik

$\cos x < -\frac{\sqrt{2}}{2}$

tengsizlikni yeching.

• Birlik aylanada $x = -\dfrac{\sqrt{2}}{2}$ vertikal chiziq o'tkazamiz.
• Kesishish burchaklari: $x_1 = \arccos\left(-\dfrac{\sqrt{2}}{2}\right) = \dfrac{3\pi}{4}$ va $x_2 = 2\pi - \dfrac{3\pi}{4} = \dfrac{5\pi}{4}$.
• $\cos x < -\dfrac{\sqrt{2}}{2}$ bo'lgani uchun chiziqdan chapdagi yoy olinadi: $\frac{3\pi}{4} + 2\pi k < x < \frac{5\pi}{4} + 2\pi k, \quad k \in \mathbb{Z}.$

Javob: $\left(\dfrac{3\pi}{4} + 2\pi k; \; \dfrac{5\pi}{4} + 2\pi k\right), \quad k \in \mathbb{Z}$.

3-misol. Tangensli tengsizlik

$\operatorname{tg} x \le \sqrt{3}$

tengsizlikni yeching.

• Asosiy davr oralig'i: $\left(-\dfrac{\pi}{2}; \dfrac{\pi}{2}\right)$.
• $\operatorname{arctg}\sqrt{3} = \dfrac{\pi}{3}$. Tangens ushbu oraliqda o'suvchi bo'lgani uchun: $-\frac{\pi}{2} < x \le \frac{\pi}{3}.$
• $\pi k$ davr qo'shamiz: $-\frac{\pi}{2} + \pi k < x \le \frac{\pi}{3} + \pi k, \quad k \in \mathbb{Z}.$

Javob: $\left(-\dfrac{\pi}{2} + \pi k; \; \dfrac{\pi}{3} + \pi k\right], \quad k \in \mathbb{Z}$.

4-misol. Kvadrat tengsizlikka keltiriladigan masala

$2\cos^2 x - \cos x - 1 \le 0$

tengsizlikning $[0; 2\pi]$ kesmadagi barcha yechimlarini toping.

• $t = \cos x$ desak: $2t^2 - t - 1 \le 0 \implies (2t + 1)(t - 1) \le 0 \implies -\dfrac{1}{2} \le t \le 1$.
• $\cos x \le 1$ barcha $x$ larda doimo bajariladi. Demak, $\cos x \ge -\dfrac{1}{2}$ tengsizlikni yechish kifoya.
• $[0; 2\pi]$ da $\cos x = -\dfrac{1}{2}$ nuqtalar: $x_1 = \dfrac{2\pi}{3}$ va $x_2 = \dfrac{4\pi}{3}$.
• $x = -0{,}5$ chiziqdan o'ngdagi yoylar olinadi: $x \in \left[0; \frac{2\pi}{3}\right] \cup \left[\frac{4\pi}{3}; 2\pi\right].$

Javob: $\left[0; \dfrac{2\pi}{3}\right] \cup \left[\dfrac{4\pi}{3}; 2\pi\right]$.

Mashq

Quyidagi topshiriqlarni mustaqil bajaring:

1. $\sin x > \dfrac{\sqrt{3}}{2}$ tengsizlikning umumiy yechimini yozing.
2. $\cos x \ge \dfrac{1}{2}$ tengsizlikning umumiy yechimini toping.
3. $\cos x < 0$ tengsizlikning $[0; 2\pi]$ kesmadagi yechimini aniqlang.
4. $\operatorname{tg} x \ge 1$ tengsizlikni yeching.
5. $\sin x \le -\dfrac{1}{2}$ tengsizlikning umumiy yechimini toping.
6. $\sin 2x > 0$ tengsizlikni yeching.
7. $\cos x > -1$ tengsizlikning umumiy yechimini yozing.
8. $\operatorname{tg} x < -\sqrt{3}$ tengsizlikni yeching.
9. $\sqrt{3} - 2\cos x > 0$ tengsizlikning $[0; \pi]$ kesmadagi yechimini toping.
10. $2\sin^2 x - 5\sin x + 2 > 0$ tengsizlikni yeching.

Javoblar va yechim yo'llari

1. Yuqori yoy: $\dfrac{\pi}{3} + 2\pi k < x < \dfrac{2\pi}{3} + 2\pi k$. Javob: $\left(\dfrac{\pi}{3} + 2\pi k; \; \dfrac{2\pi}{3} + 2\pi k\right), \; k \in \mathbb{Z}$.

2. O'ng yoy: $-\dfrac{\pi}{3} + 2\pi k \le x \le \dfrac{\pi}{3} + 2\pi k$. Javob: $\left[-\dfrac{\pi}{3} + 2\pi k; \; \dfrac{\pi}{3} + 2\pi k\right], \; k \in \mathbb{Z}$.

3. $\cos x = 0$ nuqtalar orasidagi chap yoy: $\left(\dfrac{\pi}{2}; \dfrac{3\pi}{2}\right)$. Javob: $\left(\dfrac{\pi}{2}; \dfrac{3\pi}{2}\right)$.

4. Tangens o'suvchi: $\dfrac{\pi}{4} + \pi k \le x < \dfrac{\pi}{2} + \pi k$. Javob: $\left[\dfrac{\pi}{4} + \pi k; \; \dfrac{\pi}{2} + \pi k\right), \; k \in \mathbb{Z}$.

5. Pastki yoy: $-\dfrac{5\pi}{6} + 2\pi k \le x \le -\dfrac{\pi}{6} + 2\pi k$. Javob: $\left[-\dfrac{5\pi}{6} + 2\pi k; \; -\dfrac{\pi}{6} + 2\pi k\right], \; k \in \mathbb{Z}$.

6. $2\pi k < 2x < \pi + 2\pi k \implies \pi k < x < \dfrac{\pi}{2} + \pi k$. Javob: $\left(\pi k; \; \dfrac{\pi}{2} + \pi k\right), \; k \in \mathbb{Z}$.

7. $\cos x = -1$ dan boshqa barcha nuqtalar: $x \neq \pi + 2\pi k$. Javob: $x \neq \pi + 2\pi k, \; k \in \mathbb{Z}$.

8. Tangens davrida: $-\dfrac{\pi}{2} + \pi k < x < -\dfrac{\pi}{3} + \pi k$. Javob: $\left(-\dfrac{\pi}{2} + \pi k; \; -\dfrac{\pi}{3} + \pi k\right), \; k \in \mathbb{Z}$.

9. $\cos x < \dfrac{\sqrt{3}}{2}$. $[0; \pi]$ kesmada: $\left(\dfrac{\pi}{6}; \pi\right]$. Javob: $\left(\dfrac{\pi}{6}; \pi\right]$.

10. $(2t - 1)(t - 2) > 0 \implies \sin x < \dfrac{1}{2}$ ($\sin x > 2$ bo'sh to'plam). Javob: $\left(-\dfrac{7\pi}{6} + 2\pi k; \; \dfrac{\pi}{6} + 2\pi k\right), \; k \in \mathbb{Z}$.

Keng tarqalgan xatolar

1. Tangens asimptotasini hisobga olmaslik. ✗ $\operatorname{tg} x \ge 1$ yechimini $\left[\dfrac{\pi}{4} + \pi k; \; \pi + \pi k\right)$ deb yozish. ✓ Tangens $x = \dfrac{\pi}{2} + \pi k$ nuqtalarda mavjud emas, shuning uchun oraliq $\dfrac{\pi}{2} + \pi k$ da uziladi.

2. Aylanada burchak tartibini buzish. ✗ $\sin x \ge \dfrac{1}{2}$ yechimini $\left[\dfrac{5\pi}{6}; \; \dfrac{\pi}{6}\right]$ deb teskari yozish. ✓ Burchaklar soat miliga teskari yo'nalishda kichigidan kattasiga qarab yoziladi: $\left[\dfrac{\pi}{6}; \; \dfrac{5\pi}{6}\right]$.

3. Davrni noto'g'ri tanlash. ✗ Sinus va kosinus tengsizliklariga $\pi k$ davr qo'shish. ✓ Sinus va kosinusga $2\pi k$, tangens va kotangensga $\pi k$ qo'shiladi.

4. Chegaraviy hollarni tekshirmaslik. ✗ $\cos x > 1{,}2$ tengsizlikni yechishga urinish. ✓ $-1 \le \cos x \le 1$ bo'lgani sababli bunday tengsizlik yechimga ega emas ($\emptyset$).

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