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Maxsus burchaklarning trigonometrik nisbatlari

45-45-90 uchburchaklarning va shuningdek, 30-60-90 uchburchaklarning sinusi, kosinusi va tangensini topishni oʻrganing.
Hozirgacha burchak sinusi, kosinusi va tangensini hisoblash uchun kalkulyatordan foydalanib kelganmiz. Muayyan burchakni topish uchun trigonometrik funksiyalarni kalkulyator yordamisiz ham hisoblashning iloji bor.
Gap shundaki, tomonlari nisbati bizga maʼlum boʻlgan ikkita maxsus uchburchak bor. Bular 45-45-90 uchburchak va 30-60-90 uchburchaklardir.

Maxsus uchburchaklar

30-60-90 uchburchaklar
30-60-90 uchburchak burchaklari 30, degrees va 60, degrees gradus boʻlgan toʻgʻri burchakli uchburchakdir.
45-45-90 uchburchaklar
45-45-90 uchburchak– bu burchaklari 45, degrees gradus boʻlgan toʻgʻri burchakli uchburchak.

30, degrees burchakning trigonometrik nisbati

Endi biz bu maxsus uchburchaklarning trigonometrik funksiyalarini hisoblashga tayyormiz. Keling, 30, degrees dan boshlaymiz.
Quyidagi yechilgan misolni oʻrganib, bu qanday amalga oshirilganini kuzating.

sine, left parenthesis, 30, degrees, right parenthesis nimaga teng?

Bu – yechilgan misol:
1-qadam: Bizni qiziqtirgan burchakni oʻz ichiga oluvchi maxsus uchburchakni chizing.
2-qadam: Uchburchakning tomonlarini ushbu maxsus uchburchak nisbatlariga koʻra belgilang.
3-qadam: Trigonometrik funksiya taʼrifidan foydalanib, koʻrsatilgan ifoda qiymatini toping.
sin(30)=qarshisidagi katet gipotenuza=x2x=1x2x=12\begin{aligned} \sin (30^\circ) &= \dfrac{\text{qarshisidagi katet }}{\text{gipotenuza}} \\\\ &= \dfrac{x}{2x} \\\\ &= \dfrac{1\maroonD{\cancel{x}}}{2\maroonD{\cancel{x}}} \\\\ &=\dfrac{1}{2}\end{aligned}
Diqqqat qiling, x had 1, x ga teng, shu bois start fraction, x, divided by, 2, x, end fraction, equals, start fraction, 1, x, divided by, 2, x, end fraction, equals, start fraction, 1, divided by, 2, end fraction boʻlishi tushunarli.
Endi bu usuldan foydalanib cosine, left parenthesis, 30, degrees, right parenthesis va tangent, left parenthesis, 30, degrees, right parenthesis burchaklarni topamiz.

cosine, left parenthesis, 30, degrees, right parenthesis nimaga teng?

tangent, left parenthesis, 30, degrees, right parenthesis nimaga teng?

45, degrees burchakning trigonometrik nisbati

Keling, endi ushbu jarayonni 45, degrees bilan sinab koʻramiz. Bu safar 45-45-90 uchburchak chizib, tomonlarini belgilashdan boshlaymiz.

cosine, left parenthesis, 45, degrees, right parenthesis nimaga teng?

sine, left parenthesis, 45, degrees, right parenthesis nimaga teng?

tangent, left parenthesis, 45, degrees, right parenthesis nimaga teng?

60degrees burchakning trigonometrik nisbatlari

30, degrees, 45, degrees va 60, degrees maxsus burchaklar uchun trigonometrik funksiyani hosil qilish jarayoni bir xil.
60, degrees burchakning trigonometrik nisbatini topishni batafsil koʻrsatmagan boʻlsak-da, bizga zarur barcha maʼlumotga egamiz!

cosine, left parenthesis, 60, degrees, right parenthesis nimaga teng?

sine, left parenthesis, 60, degrees, right parenthesis nimaga teng?

tangent, left parenthesis, 60, degrees, right parenthesis nimaga teng?

Xulosa

Biz 30, degrees, 45, degrees va 60, degrees burchklar uchun trigonometrik funksiyani hisoblab chiqdik. Quyidagi jadvalda erishgan natijalarimiz jamlangan.
cosine, left parenthesis, theta, right parenthesissine, left parenthesis, theta, right parenthesistangent, left parenthesis, theta, right parenthesis
theta, equals, 30, degreesstart color #1fab54, start fraction, square root of, 3, end square root, divided by, 2, end fraction, end color #1fab54start color #1fab54, start fraction, 1, divided by, 2, end fraction, end color #1fab54start color #1fab54, start fraction, square root of, 3, end square root, divided by, 3, end fraction, equals, start fraction, 1, divided by, square root of, 3, end square root, end fraction, end color #1fab54
theta, equals, 45, degreesstart color #aa87ff, start fraction, square root of, 2, end square root, divided by, 2, end fraction, equals, start fraction, 1, divided by, square root of, 2, end square root, end fraction, end color #aa87ffstart color #aa87ff, start fraction, square root of, 2, end square root, divided by, 2, end fraction, equals, start fraction, 1, divided by, square root of, 2, end square root, end fraction, end color #aa87ffstart color #aa87ff, 1, end color #aa87ff
theta, equals, 60, degreesstart color #1fab54, start fraction, 1, divided by, 2, end fraction, end color #1fab54start color #1fab54, start fraction, square root of, 3, end square root, divided by, 2, end fraction, end color #1fab54start color #1fab54, square root of, 3, end square root, end color #1fab54
Ushbu qiymatlar koʻpincha murakkab trigonometriya misollarida uchraydi. Shu sababli ularni bilish foydadan xoli emas.
Baʼzi odamlar ushbu qiymatlarni yodlashni maʼqul koʻradi, ammo yodlash shart emas. Ushbu qoʻllanmada siz qiymatlarni oʻzingiz aniqladingiz, kelajakda kerak boʻlganda ularni qayta hosil qila olasiz deb oʻylayman.