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Azizjon B.K.
Qo 'qon davlat pedagogika instituti talabasi
MURAKKAB ARGUMENTLI TRIGONOMETRIK TENGLAMALARNI
YECHISH
Annotation. There are multiple standard and non-standard methods for solving trigonometric equations, and these methods can vary significantly. Solving trigonometric equations with complex arguments often presents a greater challenge for students. Even when they manage to solve the equation, they may struggle to identify the key elements of the solution. Addressing these challenges by incorporating the study of solving such equations and inequalities into both mathematics circles andfaculty training is crucial.
Keywords: trigonometric equations, complex arguments, lowering the degree.
Azizjon B.K. student
Kokand State Pedagogical Institute
SOLVING TRIGONOMETRIC EQUATIONS WITH COMPLEX
ARGUMENTS
Annotation. There are multiple standard and non-standard methods for solving trigonometric equations, and these methods can vary significantly. Solving trigonometric equations with complex arguments often presents a greater challenge for students. Even when they manage to solve the equation, they may struggle to identify the key elements of the solution. Addressing these challenges by incorporating the study of solving such equations and inequalities into both mathematics circles andfaculty training is crucial.
Keywords: trigonometric equations, complex arguments, lowering the degree.
1-misol. Ushbu
sin(sin(cos x - sin x)) = 0
tenglamani yeching. [l]
Yechish: sin(cosx - sinx) = 7n, n g Z. -1 < sin t < 1 bo'lgani uchun -1 <7n < 1 bo'lishi kerak. Bundan n = 0. Endi sin(cosx - sinx) = 0tenglamani yechamiz. Bu cos x - sin x = 7k, k g Z tenglamaga teng kuchli.
sin(x) = *^2—к . к ning qiymati -l<—= < l tengsizlikni
ж
qanoatlantirishi kerak, bundan к = О. Demak, sin(x - — ) = О tenglamani yechamiz. x = — + —m, m e Z
Javob: — + —m,m e Z. 4
2-misol. Ushbu
cos(—Vl - sin x ) = l
tenglamani yeching. [l]
Yechish: V'l - sin x = 2к, к e N0 bo'lishi keгak.U holda sinx = l - 4к2,
-l < sin x < l bo'lgani uchun -l < l - 4к2 < l, -2 < -4к2 < О; О < к2 < l.
Demak, к = О va sinx = l.
Javob:— + —n, n e Z. 2
3-misol. Ushbu
sin2(l - cos x) = cos2(l + cos x) tenglamani yeching. [2]
Yechish. Darajani pasaytirish foгmulasini qo'llab
cos(2 + 2 cos x) + cos(2 - 2 cos x) = О tenglamani hosil qilamiz. Kosinus^ yig'indisini ko'paytmaga o'tkazib
cos(2cosx) = О; cosx = —+ —n, n e Z ni hosil qilamiz. n ning qabul qilishi
ж —n
mumkin bo'lgan qiymatlaгini topish uchun -l < — + — < l tengsizlikni
i i —
yechamiz. Bundan n = О yoki n = - l. Demak, |cos x| = —
—
Javob: + arccos —+ —к, к e Z. 4
4-misol. Ushbu
cos x + cosVx = 2
tenglamani yeching. [l]
Yechish: \cosx\ < l va |co^Vx < l bo'lgani uchun berilgan tenglama quyidagi sistemaga teng kuchli
cosx = 1 Г x = 2лк, к е Z cos Vx = 1 ' [x = 4л2m2,m е Z
Bundan 2лк = 4л2m2;к = 2лт2, к va m lar butun son bo'lganligi uchun oxirgi tenglik к = m = 0 da o'rinli.
Javob:0
5-misol. Ushbu
cos Vx = cosV x +1
tenglamani yeching. [2]
Yechish: Berilgan tenglama quyidagi tenglamaga teng kuchli. . Vx - Vx + 1 Vx + Vx + 1 л
- 2 sin-sin-= 0
2 2
. Vx+Г-yfx . Vx +V x + 1 . Bundan sin---= 0 ; sin---= 0. x ni topish uchun
quyidagi 2 ta tenglamani yechamiz. Vx +1 -л/x = 2лк, к е N; •ч/x + V x +1 = 2лп,п е N;
x > 0 da Vx +1 - Vx > 0 bo'lganligi uchun к е N
x > 0 da Vx + Vx +1 > 0 bo'lganligi uchun к е N.
Birinchi tenglamani yechamiz. x +1 = x + 4лky¡x + 4л:2 к2 ;
4лк^л = 1 - 4л2 к2
Bu tenglama к е N da yechimga ega emas.Ikkinchi tenglamani yechamiz. Vx = 2лп- V x +1. Bu tenglama quyidagi sistemaga teng kuchli.
x = 4л2n2 - 4лnVx +1 + x +1; 2лп > vx +1
Sistemaning birinchi tenglamasidan Vx +1 = 4л n +1
4лп
Vx +1 < 2лп,(п е N ) ekanligini ko'rsatamiz.Haqiqatan ham,
4л2 n2 +1 „ -4л2 n2 +1 л
-< 2лп; -< 0
4лп 4лп
Oxirgi tengsizlik n е N da o'rinli.
__ , ,4л2п2 +V2 ,4Л2П2 -1.2
Demak, x = (-)2 -1 = (-)2, n е N
4лп 4лп
. ,4л2п2 -12
Javob:, (-)2, n е N
4лп
Foydalanilgan adabiyotlar:
1. Севрюков П.Ф., Смоляков А.Н. Тригонометрические, показательные и логарифмические уравнения и неравенства. Москва.Ставрополь 2008.
2. Моденов В.П Математика. Пособие для поступающих в вузы. Москва. Новая волна. 2002.
