21-§. Burchakning sinusi,kosinusi,tangensi va kotangensi ta'riflari



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21-§. BURCHAKNING SINUSI,KOSINUSI,TANGENSI VA KOTANGENSI TA'RIFLARI 

 Mavzu

BURCHAKNING SINUSI,KOSINUSI,TANGENSI VA KOTANGENSI TA'RIFLARI .

Maqsad va vazifalar

Darsning maqsadi: o’quvchilarga Burchakning sinusi, kosinusi, tangensi va kotangensi ta’riflari haqida ma`lumot berish.

Darsning ta’limiy vazifasi: o‘quvchilarga Burchakning sinusi, kosinusi, tangensi va kotangensi ta’riflari haqida bilim berish va ulardan foydalanish ko’nikmasini hosil qilish. Darsning tarbiyaviy vazifasi: o’quvchilarni yangi bilimlar egallashga va tartib-intizomga doimo rioya etishga hamda o`qituvchilik kasbiga yo’naltirish. Darsning rivojlantiruvchi vazifasi: o‘quvchilarning kompyuterdan foydalanish haqidagi bilim va tasavvurlarini kengaytirish.

O‘quv jarayonining mazmuni

Burchakning sinusi, kosinusi, tangensi va kotangensi ta’riflarining xossalari.

O‘quv jarayonini amalga oshirish texnologiyasi

Uslub: Aralash

Shakl: Savol-javob. Jamoa va kichik gruhlarda ishlash.

Vosita: Elektron resurslar, darslik, plakatlar; tarqatma materiallar.

Usul: Tayyor prezentatsiya va slayd materiallari asosida.

Nazorat: Og‘zaki, savol-javob, muhokama, kuzatish.

Baholash: Rag‘batlantirish, 5 ballik reyting tizimi asosida.

Kutiladigan natijalar

O’quvchilar yangi bilim va ko’nikmaga ega bo’ladi. Ularga Burchakning sinusi, kosinusi, tangensi va kotangensi ta’riflari xossalari. haqida bilimga va foydalanish ko`nikmalariga ega bo`ladilar.

Kelgusi rejalar (tahlil, o‘zgarishlar)

O’qituvchi o’z faoliyatining tahlili asosida yoki hamkasblarining dars tahlili asosida keyingi darslariga o‘zgartirishlar kiritadi va rejalashtiradi.

Yangi mavzu bayoni:

Geometriya kursida graduslarda ifodalangan burchakning sinusi, kosinusi va tangensi kiritilgan edi. Bu burchak 0° dan 180° gacha bo'lgan oraliqda qaralgan. Ixtiyoriy burchakning sinusi va kosinusi quyidagicha ta'riflanadi:



1- ta`rif

a burchakning sinusi deb, (1; 0) nuqtani koordinatalar boshi atrofida a burchakka burish natijasida hosil bo'lgan nuqtaning ordinatasiga aytiladi (sina kabi belgilanadi). c:\program files (x86)\imt\algebra 9-sinf\data\res_style\rb.gif

2- ta`rif

a burchakning коsinusi deb, (1; 0) nuqtani koordinatalar boshi atrofida a burchakka burish natijasida hosil bo'lgan nuqtaning abssissasiga aytiladi (cosa kabi belgilanadi).

c:\program files (x86)\imt\algebra 9-sinf\data\images\img2\58 копия.gif

Bu ta'riflarda a burchak graduslarda, shuningdek, radianlarda ham ifodalanishi mumkin.

Masalan, (1; 0) nuqtani c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2534.gif burchakka, ya'ni 90° ga burishda (0; 1) nuqta hosil qilinadi.    (0; 1) nuqtaning ordinatasi 1 ga teng, shuning uchun

c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2536.gif;

bu nuqtaning abssissasi 0 ga teng, shuning uchun



c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2538.gif.

Burchak 0° dan 180° gacha oraliqda bo'lgan holda sinus va kosinuslarning ta'riflari geometriya kursidan ma'lum bo'lgan sinus va kosinus ta'riflari bilan mos tushishini ta'kidlaymiz.

Masalan,

c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2540.gif

1-masala. sin(-p) va cos(-p) ni toping.

D (1; 0) nuqtani -p burchakka burganda u (-1; 0) nuqtaga o'tadi (58- rasm).

c:\program files (x86)\imt\algebra 9-sinf\data\images\img2\59 копия.gif

58-rasm.

Shuning uchun sin(-p) = 0, cos(-p) = -1.  c:\program files (x86)\imt\algebra 9-sinf\data\images\img8\rasm-343.jpg



2-masala. sin270° va cos270° ni toping.

D (1;0) nuqtani 270° ga burganda, u (0;-1) nuqtaga o'tadi (59- rasm).

c:\program files (x86)\imt\algebra 9-sinf\data\images\img23\rasm-5.gif

59-rasm.

Shuning uchun cos 270°= 0, sin270°=-l. c:\program files (x86)\imt\algebra 9-sinf\data\images\img8\rasm-344.jpg



3-masala. sin t = 0 tenglamani yeching.

D sint = 0 tenglamani yechish - bu sinusi nolga teng bo'lgan barcha burchaklarni topish demakdir.

Birlik aylanada ordinatasi nolga teng bo'lgan ikkita nuqta bor: (1;0) va (-1; 0) (58- rasm). Bu nuqtalar (1; 0) nuqtani 0, c:\program files (x86)\imt\algebra 9-sinf\data\images\img18\eq-322008605.gif, c:\program files (x86)\imt\algebra 9-sinf\data\images\img18\eq-30328470.gif, c:\program files (x86)\imt\algebra 9-sinf\data\images\img18\eq-270760688.gif va hokazo, shuningdek, c:\program files (x86)\imt\algebra 9-sinf\data\images\img18\eq-2073930420.gif, c:\program files (x86)\imt\algebra 9-sinf\data\images\img18\eq-935717771.gif, c:\program files (x86)\imt\algebra 9-sinf\data\images\img18\eq-1622264251.gif  va hokazo burchaklarga burish bilan hosil qilinadi.

Demak, c:\program files (x86)\imt\algebra 9-sinf\data\images\img29\eq-1585766452.gif bo'lganda (bunda k - istalgan butun son) sint=0 bo'ladi. c:\program files (x86)\imt\algebra 9-sinf\data\images\img8\rasm-345.jpg

Butun sonlar to'plami Z harfi bilan belgilanadi. k son Z ga tegishli ekanligini belgilash uchunc:\program files (x86)\imt\algebra 9-sinf\data\images\img29\eq-165672715.gif yozuvdan foydalaniladi ("k son ga tegishli" deb o'qiladi). Shuning uchun 3-masala javobini bunday yozish mumkin:



c:\program files (x86)\imt\algebra 9-sinf\data\images\img29\eq-15028848.gif.

4-masala. cost = 0 tenglamani yeching.

D Birlik aylanada abssissasi nolga teng bo'lgan ikkita nuqta bor: (0, 1) va (0; -1) (60- rasm).

c:\program files (x86)\imt\algebra 9-sinf\data\images\img2\60 копия.gif

60- rasm.

Bu nuqtalar (1; 0) nuqtani c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2542.gif va hokazo, shuningdek,  c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2544.gif va hokazo burchaklarga, ya'ni c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2546.gif (bunda c:\program files (x86)\imt\algebra 9-sinf\data\images\img23\eq-1774403208.gif) burchaklarga burish bilan hosil qilinadi.

Javob: c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2548.gif.  c:\program files (x86)\imt\algebra 9-sinf\data\images\img8\rasm-346.jpg7



5-masala. Tenglamani yeching: 1) sint = l; 2) cost = l.

D 1) Birlik aylananing (0; 1) nuqtasi birga teng ordinataga ega. Bu nuqta (1; 0) nuqtani c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2550.gif burchakka burish bilan hosil qilinadi.

2) (1; 0) nuqtani c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2552.gif burchakka burish bilan hosil qilingan nuqtaning abssissasi birga teng bo'ladi.

Javob:  c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2554.gif bo'lganda, sint = 1,

   c:\program files (x86)\imt\algebra 9-sinf\data\images\img29\eq-1159856955.gif bo'lganda, cost =1, c:\program files (x86)\imt\algebra 9-sinf\data\images\img18\eq-1850009962.gif.  c:\program files (x86)\imt\algebra 9-sinf\data\images\img8\rasm-347.jpg



3- ta`rif

a burchakning tangensi, deb a burchak sinusining uning kosinusiga nisbatiga aytiladi (tga kabi belgilanadi). c:\program files (x86)\imt\algebra 9-sinf\data\res_style\rb.gif

Shunday qilib,        c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2556.gif

Masalan, c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2558.gif   c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2560.gif

Ba'zan a burchakning kotangensidan foydalaniladi (ctga kabi belgilanadi). U c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2562.gif formula bilan aniqlanadi.

Masalan, c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2564.gif   c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2566.gif

sina va cosa lar ixtiyoriy burchak uchun ta'riflanganligini, ularning qiymatlari esa -1 dan 1 gacha oraliqda ekanligini ta'kidlab o'tamiz; c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2568.gif faqat c:\program files (x86)\imt\algebra 9-sinf\data\images\img23\eq-1679448431.gif bo'lgan burchaklar uchun, ya'ni c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2570.gif dan boshqa ixtiyoriy burchaklar uchun aniqlangan.



Sinus, kosinus, tangens va kotangenslarning ko'proq uchrab turadigan qiymatlari jadvalini keltiramiz.   

c:\program files (x86)\imt\algebra 9-sinf\data\images\img23\eq-1542656510.gif

0

c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2559.gif

c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2561.gif

c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2563.gif

c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2565.gif

c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2567.gif

c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2569.gif

c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2571.gif

(0°)

(30°)

(45°)

(60°)

(90°)

(180°)

(270°)

(360°)

c:\program files (x86)\imt\algebra 9-sinf\data\images\img23\eq-694054902.gif

0

c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2573.gif

c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2575.gif

c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2577.gif

1

0

-1

0

c:\program files (x86)\imt\algebra 9-sinf\data\images\img23\eq-503067925.gif

1

c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2579.gif

c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2581.gif

c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2583.gif



-1

0

1

c:\program files (x86)\imt\algebra 9-sinf\data\images\img23\eq-426528028.gif

0

c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2585.gif

1

c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2587.gif

Mavjud emas

0

Mavjud emas

0

c:\program files (x86)\imt\algebra 9-sinf\data\images\img23\eq-1910685130.gif

Mavjud emas

c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2589.gif

1

c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2591.gif

0

Mavjud emas

0

Mavjud emas

6-masala. Hisoblang:

c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2572.gif

D Jadvaldan foydalanib, hosil qilamiz:

c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2574.gif c:\program files (x86)\imt\algebra 9-sinf\data\images\img9\rasm-348.jpg

Sinus, kosinus, tangens va kotangenslarning bu jadvalga kirmagan burchaklar uchun qiymatlarini V.M.Bradisning to'rt xonali matematik jadvallaridan, shuningdek, mikrokalkulator yordamida topish mumkin.

Agar har bir haqiqiy songa sinx son mos keltirilsa, u holda haqiqiy sonlar to'plamida y=sinx funksiya berilgan bo'ladi.            

Shunga o'xshash, y=cosx, y=tgx va y=ctgx funksiyalar beriladi.  y=cosx  funksiya barcha  xIR  da aniqlangany=tgx funksiyac:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2576.gif , y=ctgx esa c:\program files (x86)\imt\algebra 9-sinf\data\images\img6\rasm-2578.gif bo'lganda aniqlangan.  y=sinx  va y=cosx funksiyalarning grafiklari 61 va 62- rasmlarda tasvirlangan.



c:\program files (x86)\imt\algebra 9-sinf\data\imgdesign\1\a_ копия.gif

y=sinxy=cosx, y=tgx va y=ctgx funksiyalar trigonometrik funksiyalar deyiladi.

c:\program files (x86)\imt\algebra 9-sinf\data\imgdesign\1\b копия.gif

Darsni yakunlash.

1) faol qatnashgan o`quvchilarni baholash va rag`batlantirish.



2) Uyga vazifa berish. Mavzuga oid misollarni yechib kelish.
O`quv tarbiyaviy ishlar bo`yicha direktor o`rinbosari S. Axmedov



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