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Trigonometriya

Course: Trigonometriya > Unit 1

Lesson 9: Teskari trigonometrik nisbatlar

Teskari trigonometrik funksiyalar

Bilib oling: kosekans, sekans va kotangenslar asosiy trigonometrik nisbatlarning (sinus, kosinus va tangenslarning) teskarisidir.

Biz allaqachon asosiy trigonometrik nisbatlarni oʻrgandik:

Hali oʻrganish uchun yana uchta nisbat bor:

start fraction, start color #11accd, a, end color #11accd, divided by, start color #aa87ff, c, end color #aa87ff, end fraction oʻrniga start fraction, start color #aa87ff, c, end color #aa87ff, divided by, start color #11accd, a, end color #11accd, end fraction deb olishimiz mumkin.
start fraction, start color #ed5fa6, b, end color #ed5fa6, divided by, start color #aa87ff, c, end color #aa87ff, end fraction oʻrniga start fraction, start color #aa87ff, c, end color #aa87ff, divided by, start color #ed5fa6, b, end color #ed5fa6, end fraction deb olishimiz mumkin.
start fraction, start color #11accd, a, end color #11accd, divided by, start color #ed5fa6, b, end color #ed5fa6, end fraction oʻrniga start fraction, start color #ed5fa6, b, end color #ed5fa6, divided by, start color #11accd, a, end color #11accd, end fraction deb olishimiz mumkin.

Bu yangi nisbatlar oʻzaro trigonometrik nisbatlar boʻlib, ularning nomlari bilan tanishamiz.

Kosekans left parenthesis, \csc, right parenthesis

Kosekans sinusning aksidir. U toʻgʻri burchakli uchburchakda gipotenuzaning berilgan burchak qarshisidagi katetga nisbatiga teng.

sine, left parenthesis, A, right parenthesis, equals, start fraction, start color #11accd, start text, q, a, r, s, h, i, s, i, d, a, g, i, space, k, a, t, e, t, end text, end color #11accd, divided by, start color #aa87ff, start text, g, i, p, o, t, e, n, u, z, a, end text, end color #aa87ff, end fraction, equals, start fraction, start color #11accd, a, end color #11accd, divided by, start color #aa87ff, c, end color #aa87ff, end fraction

\csc, left parenthesis, A, right parenthesis, equals, start fraction, start color #aa87ff, start text, g, i, p, o, t, e, n, u, z, a, end text, end color #aa87ff, divided by, start color #11accd, start text, q, a, r, s, h, i, s, i, d, a, g, i, space, k, a, t, e, t, end text, end color #11accd, end fraction, equals, start fraction, start color #aa87ff, c, end color #aa87ff, divided by, start color #11accd, a, end color #11accd, end fraction

Sekans left parenthesis, \sec, right parenthesis

Sekans kosinusning aksidir. U toʻgʻri burchakli uchburchakda gipotenuzaning berilgan burchakka yopishgan katetiga nisbatiga teng.

cosine, left parenthesis, A, right parenthesis, equals, start fraction, start color #ed5fa6, start text, y, o, p, i, s, h, g, a, n, space, k, a, t, e, t, end text, end color #ed5fa6, divided by, start color #aa87ff, start text, g, i, p, o, t, e, n, u, z, a, end text, end color #aa87ff, end fraction, equals, start fraction, start color #ed5fa6, b, end color #ed5fa6, divided by, start color #aa87ff, c, end color #aa87ff, end fraction

\sec, left parenthesis, A, right parenthesis, equals, start fraction, start color #aa87ff, start text, g, i, p, o, t, e, n, u, z, a, end text, end color #aa87ff, divided by, start color #ed5fa6, start text, y, o, p, i, s, h, g, a, n, space, k, a, t, e, t, end text, end color #ed5fa6, end fraction, equals, start fraction, start color #aa87ff, c, end color #aa87ff, divided by, start color #ed5fa6, b, end color #ed5fa6, end fraction

Kotangens left parenthesis, cotangent, right parenthesis

Kotangens tangensning aksidir. U toʻgʻri burchakli uchburchakda burchakka yopishgan katetning qarshisidagi katetga nisbatiga teng.

tangent, left parenthesis, A, right parenthesis, equals, start fraction, start color #11accd, start text, q, a, r, s, h, i, s, i, d, a, g, i, space, k, a, t, e, t, end text, end color #11accd, divided by, start color #ed5fa6, start text, y, o, p, i, s, h, g, a, n, space, k, a, t, e, t, end text, end color #ed5fa6, end fraction, equals, start fraction, start color #11accd, a, end color #11accd, divided by, start color #ed5fa6, b, end color #ed5fa6, end fraction

cotangent, left parenthesis, A, right parenthesis, equals, start fraction, start color #ed5fa6, start text, y, o, p, i, s, h, g, a, n, space, k, a, t, e, t, end text, end color #ed5fa6, divided by, start color #11accd, start text, q, a, r, s, h, i, s, i, d, a, g, i, space, k, a, t, e, t, end text, end color #11accd, end fraction, equals, start fraction, start color #ed5fa6, b, end color #ed5fa6, divided by, start color #11accd, a, end color #11accd, end fraction

Odamlar bunday narsani qanday eslab qolishadi?

Aksariyat odamlar uchun bu yangi nisbatlarni eslab qolishning oson yoʻli mazkur nisbatlarni ularga teskari nisbatlar bilan bogʻlashdir. Quyidagi jadvalda oʻsha munosabatlar aks etgan.

	Ogʻzaki taʼrif	Matematik munosabat
kosekans	Kosekans bu – sinusning teskarisi.	\csc, left parenthesis, A, right parenthesis, equals, start fraction, 1, divided by, sine, left parenthesis, A, right parenthesis, end fraction
sekans	Sekans bu – kosinusning aksi.	\sec, left parenthesis, A, right parenthesis, equals, start fraction, 1, divided by, cosine, left parenthesis, A, right parenthesis, end fraction
kotangens	Kotangens bu – tangensning teskarisi.	cotangent, left parenthesis, A, right parenthesis, equals, start fraction, 1, divided by, tangent, left parenthesis, A, right parenthesis, end fraction

Teskari trigonometrik nisbatlarni aniqlash

Keling, misollar bilan oʻrganamiz.

Quyidagi uchburchakdan \csc, left parenthesis, C, right parenthesis, \sec, left parenthesis, C, right parenthesis va cotangent, left parenthesis, C, right parenthesis burchaklarni toping.

Yechim

Kosekantni topish

Bizga maʼlumki, kosekans – bu sinusning teskarisi.

Sinus qarshisidagi katetning gipotenuzaga nisbatiga teng ekan, kosekans esa gipotenuzaning qarshisidagi katetga nisbatiga teng.

\begin{aligned}\csc (C) &= \dfrac{\purpleC{\text{gepotenuza}}} {\blueD{\text{ qarshisidagi katet}}} \\\\ &= \dfrac{{17}}{{15}} \end{aligned}

Sekansni topish

Bizga maʼlumki, sekans – bu kosinusning teskarisi.

Burchak kosinusi unga yopishgan katetning gipotenuzaga nisbatiga teng ekan, burchak sekansi esa gipotenuzaning unga yopishgan katetga nisbatiga teng.

\begin{aligned}\sec (C) &= \dfrac{\purpleC{\text{gepotenuza}}}{\maroonC{\text{yopishgan katet}}} \\\\ &= \dfrac{{17}}{{8}} \end{aligned}

Kotangensni topish

Bizga maʼlumki, kotangens – bu tangensning teskarisi.

Burchak tangensi qarshisidagi katetning unga yopishgan katetga nisbatiga teng ekan, burchak kotangensi esa unga yopishgan katetning qarshisidagi katetga nisbatiga teng.

\begin{aligned}\cot (C) &= \dfrac{\maroonC{\text{yopishgan katet}}}{\blueD{\text{qarshisidagi katet}}} \\\\ &= \dfrac{{8}}{{15}} \end{aligned}

[Koʻproq misol koʻrishni xohlayman, iltimos.]

Oʻzingiz urinib koʻring!

1-masala

\csc, left parenthesis, X, right parenthesis, equals

2-masala

\sec, left parenthesis, W, right parenthesis, equals

3-misol

cotangent, left parenthesis, R, right parenthesis, equals

Murakkab masala

\csc, left parenthesis, 45, degrees, right parenthesis aniq qiymati nechaga teng?

Muhokamaga qoʻshilmoqchimisiz?

Kirish

Saralash:

Hozircha izohlar yoʻq.

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