TrigGraphs | MathModel.uz

2.2-bo'lim: Trig funksiyalarining grafiklari

Bu bo'limda ko'p narsa yo'q. Bu sizga oltita trig funksiyalarining grafiklarini, shuningdek, trig funksiyalari haqida bir nechta yoqimli xususiyatlarni eslatish uchun.

Muammolarga o'tishdan oldin, biz buni ko'rganimizni eslang Trig funktsiyasini baholash funksiyalarni ishga tushiruvchi bo‘limga misol bo‘la oladi davriy funktsiyalari. Bu shuni anglatadiki, biz haqiqatan ham qilishimiz kerak bo'lgan narsa funktsiyani qiymatlarning bir davri uchun grafigini tuzib, keyin grafikni takrorlashdir.

Quyidagi funksiyaning grafigini tuzing.

$y = \cos \left( x \right)$

Yechim

Haqiqatan ham 0 va $2\pi $ o'rtasida bir nechta nuqtalarni chizishdan boshqa ko'p narsa yo'q, keyin takrorlang. Esingizda bo'lsin, kosinus $2\pi $ davriga ega (5-sonli muammoga qarang). Trig funktsiyasini baholash).

Mana $ - 4\pi \le x \le 4\pi $ uchun grafik.

$The graph of $y = \cos \left( x \right)$ on $ - 4\pi \le x \le 4\pi $. It is a wave with peaks at $y = 1$ and valleys at $y = - 1$ and a period of $2\pi $, so there are four complete cycles shown. The graph has a peak at $x = 0$.$

E'tibor bering, grafik $x$ diapazonida kerak bo'lganda 4 marta takrorlanadi.

Shuni ham ta'kidlab o'tamizki, biz $x$ ning barcha qiymatlarini kosinusga qo'yishimiz mumkin (bu ko'pchilik trig funktsiyalari uchun bo'lmaydi) va shuni ham ta'kidlaymizki,
\[ - 1 \le \cos \left( x \right) \le 1\]
Shuni ta'kidlash kerakki, kosinus hech qachon 1 dan katta yoki -1 dan kichik bo'lmaydi. Bu ba'zan hisob darslarida foydali bo'ladi.
$y = \cos \left( {2x} \right)$

Yechim

Ushbu grafik bilan biroz ehtiyot bo'lishimiz kerak. $\cos \left( x \right)$ $2\pi $ davriga ega, ammo biz bu erda $\cos \left( x \right)$ bilan ishlamayapmiz. Biz $\cos \left( {2x} \right)$ bilan ishlaymiz. Bunday holda, agar biz $x = \pi $ ni ulab olsak, biz olamiz
\[\cos \left( {2\left( \pi \right)} \right) = \cos \left( {2\pi } \right) = \cos \left( 0 \right) = 1\]
Bu holda funksiya $2\pi $ o'rniga $\pi $ dan keyin takrorlana boshlaydi! Demak, bu funksiya $\pi $ davriga ega. Shunday qilib, biz grafik $ - 4\pi \le x \le 4\pi $ oralig'ida 8 marta takrorlanishini kutishimiz mumkin. Mana o'sha grafik.

$The graph of $y = \cos \left( {2x} \right)$ on $ - 4\pi \le x \le 4\pi $. It is a wave with peaks at $y = 1$ and valleys at $y = - 1$ and a period of $\pi $, so there are eight complete cycles shown, twice as many as the graph of $\cos \left( x \right)$.$

Albatta, bu grafikda ikki barobar ko'p tsikllar mavjud.

Umuman olganda, biz $\cos \left( {\omega \,x} \right)$ davrini quyidagidan foydalanib olishimiz mumkin.
\[{\rm{Period}} = \frac{{2\pi }}{\omega }\]
Agar $\omega > 1$ bo'lsa, biz $2\pi $ dan kichikroq davrni kutishimiz mumkin va shuning uchun grafik tezroq tebranadi. Xuddi shunday, agar $\omega < 1$ bo'lsa, biz $2\pi $ dan kattaroq davrni kutishimiz mumkin va shuning uchun grafik sekinroq tebranadi.

E'tibor bering, davr qancha kosinus olishiga ta'sir qilmaydi. Bizda hali ham bor
\[ - 1 \le \cos \left( {2x} \right) \le 1\]
$y = 5\cos \left( {2x} \right)$

Yechim

Bu holda men kosinus oldiga 5 qo'shdim. Buning hammasi kosinusning qanchalik katta bo'lishini oshiradi. Kosinus yoki sinus oldidagi raqam deyiladi amplituda. Mana bu funktsiyaning grafigi.

$The graph of $y = 5\cos \left( {2x} \right)$ on $ - 4\pi \le x \le 4\pi $. It is a wave with the same period of $\pi $ as the graph of $\cos \left( {2x} \right)$, so there are eight complete cycles shown, but the peaks are now at $y = 5$ and the valleys at $y = - 5$.$

Ushbu muammo uchun $y$ o'qidagi o'lchovga e'tibor bering va uni oldingi grafik bilan aralashtirib yubormang. $y$ o'qi shkalasi boshqacha!

Umuman olganda,
\[ - R \le R\cos \left( {\omega \,x} \right) \le R\]
$y = \sin \left( x \right)$

Yechim

Ushbu bo'limdagi birinchi muammoda bo'lgani kabi, uni grafik qilishdan boshqa ko'p narsa qilish kerak emas. Mana $ - 4\pi \le x \le 4\pi $ diapazonidagi grafik.

$The graph of $y = \sin \left( x \right)$ on $ - 4\pi \le x \le 4\pi $. It is a wave with peaks at $y = 1$ and valleys at $y = - 1$ and a period of $2\pi $, so there are four complete cycles shown. The graph passes through the origin going upwards.$

Ushbu grafikdan biz sinus kosinus bilan bir xil diapazonga ega ekanligini ko'rishimiz mumkin. Umuman olganda
\[ - R \le R\sin \left( {\omega \,x} \right) \le R\]
Kosinusda bo'lgani kabi, sinusning o'zi hech qachon 1 dan katta bo'lmaydi va hech qachon -1 dan kichik bo'lmaydi.
$\displaystyle y = \sin \left( {\frac{x}{3}} \right)$

Yechim

Shunday qilib, bu holda bizda qavs ichida faqat $x$ mavjud emas. Xuddi kosinus holatida bo'lgani kabi biz $\sin \left( {\omega \,x} \right)$ davrini foydalanib olishimiz mumkin
\[\text{Period}=\frac{2\pi }{\omega }=\frac{2\pi }{{}^{1}/{}_{3}}=6\pi \]
Bu holda egri chiziq har bir $6\pi $ takrorlanadi. Shunday qilib, ushbu grafik uchun diapazonni $ - 6\pi \le x \le 6\pi $ ga o'zgartiraman, shunda biz egri chiziqning kamida ikkita izini olishimiz mumkin. Mana grafik.

$The graph of $y = \sin \left( {\frac{x}{3}} \right)$ on $ - 6\pi \le x \le 6\pi $. It is a wave with peaks at $y = 1$ and valleys at $y = - 1$ and a period of $6\pi $, so only two complete cycles are shown. The graph passes through the origin going upwards.$
$y = \tan \left( x \right)$

Yechim

Tangens holatida biz $x$-ni ulashda ehtiyot bo'lishimiz kerak, chunki kosinus nolga teng bo'lgan joyda tangens mavjud emas (esda tutingki, $\tan x = \frac{{\sin x}}{{\cos x}}$). Tangent mavjud bo'lmaydi
\[x = \cdots , - \frac{{5\pi }}{2}, - \frac{{3\pi }}{2}, - \frac{\pi }{2},\frac{\pi }{2},\frac{{3\pi }}{2},\frac{{5\pi }}{2}, \ldots \]
va grafikning bu nuqtalarda asimptotalari bo'ladi. Mana $ - \frac{{5\pi }}{2} < x < \frac{{5\pi }}{2}$ diapazonidagi tangens grafigi.

$The graph of $y = \tan \left( x \right)$ on $ - \frac{{5\pi }}{2} < x < \frac{{5\pi }}{2}$. There are dashed vertical asymptotes at $x = - \frac{{5\pi }}{2}$, $ - \frac{{3\pi }}{2}$, $ - \frac{\pi }{2}$, $\frac{\pi }{2}$, $\frac{{3\pi }}{2}$ and $\frac{{5\pi }}{2}$. Between each pair of asymptotes the graph is an increasing curve that comes up from negative infinity on the left, passes through the x-axis at the midpoint and goes up to positive infinity on the right, and this shape repeats with a period of $\pi $.$

Va nihoyat, $R\tan \left( {\omega \,x} \right)$ haqida bir nechta tezkor xususiyatlar.
\[\begin{array}{c} - \infty < R\tan \left( {\omega \,x} \right) < \infty \\ {\rm{Period}} = \displaystyle \frac{\pi }{\omega }\end{array}\]
Ushbu davr uchun $\tan \left( x \right)$ sinus va kosinusdan farqli o'laroq $\pi $ davriga ega ekanligini unutmang va bu sinus va kosinus uchun hisoblagichda 2 ning yo'qligi bilan bog'liq.
$y = \sec \left( x \right)$

Yechim

Tangentda bo'lgani kabi, biz kosinusu nolga teng bo'lgan $x$ dan qochishimiz kerak (esda tutingki, $\sec x = \frac{1}{{\cos x}}$). Bu vaqtda sekant mavjud bo'lmaydi
\[x = \cdots , - \frac{{5\pi }}{2}, - \frac{{3\pi }}{2}, - \frac{\pi }{2},\frac{\pi }{2},\frac{{3\pi }}{2},\frac{{5\pi }}{2}, \ldots \]
va grafikning bu nuqtalarda asimptotalari bo'ladi. Mana $ - \frac{{5\pi }}{2} < x < \frac{{5\pi }}{2}$ diapazonidagi sekant grafigi.

$The graph of $y = \sec \left( x \right)$ on $ - \frac{{5\pi }}{2} < x < \frac{{5\pi }}{2}$. There are dashed vertical asymptotes at $x = - \frac{{5\pi }}{2}$, $ - \frac{{3\pi }}{2}$, $ - \frac{\pi }{2}$, $\frac{\pi }{2}$, $\frac{{3\pi }}{2}$ and $\frac{{5\pi }}{2}$. Between the asymptotes the graph alternates between a “U” shaped piece sitting above the x-axis whose lowest point is at $y = 1$ and an upside down “U” shaped piece below the x-axis whose highest point is at $y = - 1$. The graph never takes values between -1 and 1.$

Grafik har doim 1 dan katta yoki -1 dan kichik ekanligiga e'tibor bering. Bu juda ajablanarli bo'lmasligi kerak. Eslatib o'tamiz, $ - 1 \le \cos \left( x \right) \le 1$. Shunday qilib, 1 dan kichik bo'lgan 1 dan katta bo'ladi. Shuningdek, ${1}/{\pm 1}=\pm 1$ va shuning uchun biz sekantdan quyidagi diapazonlarni olamiz.
\[R\sec \left( {\omega \,x} \right) \ge R\hspace{0.5in}{\rm{and}}\hspace{0.55in}R\sec \left( {\omega \,x} \right) \le - R\]
$y = \csc \left( x \right)$

Yechim

Ushbu grafik uchun biz $x$ dan qochishimiz kerak, bu erda sinus nolga teng $\left( {\csc x = \frac{1}{{\sin x}}} \right)$. Demak, kosekant grafigi uchun mavjud bo'lmaydi
\[x = \cdots , - 2\pi , - \pi ,0,\pi ,2\pi , \cdots \]
Mana kosekantning grafigi.

$The graph of $y = \csc \left( x \right)$ on $ - 3\pi < x < 3\pi $. There are dashed vertical asymptotes at $x = - 3\pi $, $ - 2\pi $, $ - \pi $, $0$, $\pi $, $2\pi $ and $3\pi $. Between the asymptotes the graph alternates between a “U” shaped piece sitting above the x-axis whose lowest point is at $y = 1$ and an upside down “U” shaped piece below the x-axis whose highest point is at $y = - 1$. The graph never takes values between -1 and 1.$

Kosekant sekant bilan bir xil diapazonga ega bo'ladi.
\[R\csc \left( {\omega \,x} \right) \ge R\hspace{0.5in}{\rm{and}}\hspace{0.5in}R\csc \left( {\omega \,x} \right) \le - R\]
$y = \cot \left( x \right)$

Yechim

Kotangentdan qochish kerak
\[x = \cdots , - 2\pi , - \pi ,0,\pi ,2\pi , \cdots \]
chunki bu nuqtalarda bizda nolga bo'linish bo'ladi. Mana grafik.
$The graph of $y = \cot \left( x \right)$ on $ - 3\pi < x < 3\pi $. There are dashed vertical asymptotes at $x = - 3\pi $, $ - 2\pi $, $ - \pi $, $0$, $\pi $, $2\pi $ and $3\pi $. Between each pair of asymptotes the graph is a decreasing curve that comes down from positive infinity on the left, crosses the x-axis at the midpoint and drops to negative infinity on the right, and this shape repeats with a period of $\pi $.$
Kotangent quyidagi diapazonga ega.
\[ - \infty < R\cot \left( {\omega \,x} \right) < \infty \]