Trigonometrik tengsizliklarni yechish usullari

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Trigonometrik tengsizliklarni yechish usullari
Ushbu sinx >a, cosx>a, tgx>a, ctgx>a, sinx ≤ a, cosx ≤a, tgx≤a, ctg≤a ko’rinishdagi tengsizliklarni eng sodda trigonometrik tengsizliklar deyiladi. Murakkab tengsizliklarni yechish bu sodda tengsizliklardan birini yechishga keltiriladi.

Tengsizlikning ko’rinishi	Tengsizlikning yechimlari to’plami
sinx>a	 
sinx	 
cosx.>a	 
cosx	 
tg>a	 
tg	 
ctgx>a	 
ctgx	 
	 

2sin²x-7sinx+3 > 0 tengsizlikni yeching

2y²-7y+3 > 0

<sub> 

 </sub> tengsizlikning yechimi bo’ladi.

_{ } tengsizlikni yeching

Berilgan tengsizlikdan _{ } tengsizliklarni olamiz. Birinchi tengsizlikning yechimlari to’plami : _{ }

Ikkinchi tengsizlikning yechimlari to’plami : _{ }.

2. _{ }

3. _{ }

Trigonometrik tengsizliklarga oid formulalar ;

1. sinx ≥ a, −1 ≤ a ≤ 1

2nπ +arcsina ≤ x ≤ −arcsina +(2n+1)π, n ∈Z

2. sinx ≤ a, −1 ≤ a ≤ 1

(2n−1)π−arcsina ≤ x ≤ arcsina+2nπ, n ∈ Z

3. cosx ≥ a, −1 ≤ a ≤ 1

2nπ − arccosa ≤ x ≤ arccosa + 2nπ, n ∈ Z

4. cosx ≤ a, −1 ≤ a ≤ 1

2nπ +arccosa ≤ x ≤ −arccosa +2(n +1)π, n ∈Z

5. tgx ≥ b, arctgb + πn ≤ x < π/2+ nπ, n ∈ Z

6. tgx ≤ b, −2π+ πn < x ≤ arctgb + nπ, n ∈ Z

7. ctgx ≥ b, nπ < x ≤ arcctgb + nπ, n ∈ Z

8. ctgx ≤ b, arcctgb + nπ ≤ x < π + nπ, n ∈ Z

Masala yechish namunasi:

Ushbu y =√2sinx − 1 funksiyaning aniqlanish sohasini toping.

A)−π/6+ 2πn; π/6+ 2πn, n ∈ Z

B) [π/6+ 2πn; 5π/6+ 2πn], n ∈ Z

C)π/6+ 2πn; 5π/6+ 2πn, n ∈ Z

D) [−π/6+ 2πn; π/6+ 2πn], n ∈ Z

E) [π/3+ πn; 2π/3+ πn], n ∈ Z

Yechish: y =√2sinx − 1 funksiya 2sinx − 1 ≥ 0

bo’lganda aniqlangan. Bu tengsizlikni

sinx ≥1/2

ko’rinishda yozamiz.

Javob: π/6+ 2πn ≤ x ≤5π/6+ 2πn, n ∈ Z (B).

Mustaqil yechish uchun testlar:

1. 
   (96-9-51) Ushbu sin2x −52sinx + 1 < 0 tengsizlik
   

x (x ∈ [0; 2π]) ning qanday qiymatlarida o’rinli?

A) [0;π/6] ∪ [5π/6; 2π]

B) (π/6;5π/6)

C) (0;π/3) ∪ (2π/3; 2π]

D) [0;π/3) ∪ (2π/3; 2π]

E) ∅
2. (96-9-105) Tengsizlikni yeching.

2sin2x ≥ ctg π/4

A) [π/6+ 2πn; 5π/6+ 2πn], n ∈ Z

B) (π/12+ πn;

5π/12+ πn), n ∈ Z

C) [π/12+ πn; 5π/12+ πn], n ∈ Z

D) [π/12+ 2πn; 5π/12+ 2πn], n ∈ Z

E) [−π/3+ 2πn; π/3+ 2πn], n ∈ Z

3. (97-9-101) Tengsizlikni yeching.

sinx · cosx >√2/4

A)π/8+ 2πk < x <3π/8+ 2πk, k ∈ Z

B)π/4+ πk < x <3π/4+ πk, k ∈ Z

C)π/8+ πk < x <3π/8+ πk, k ∈ Z

D)π/8+ πk ≤ x ≤ 3π/8+ πk, k ∈ Z

E)π/6+ πk < x <5π/6+ πk, k ∈ Z

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