Trigonometrik funksiyalar: davr, qiymatlar sohasi, a·sin x + b·cos x: matematika darsi

$y=3\sin x+4\cos x$ funksiyaning eng katta qiymati $7$ emas, $5$ : sinus va kosinus bir vaqtda $1$ ga teng bo'la olmaydi. Bu darsda funksiyalarning xossalari, davrni topish, yordamchi burchak usuli va qiymatlar sohasini topishning barcha yo'llari to'planadi.

To'rt funksiyaning xossalari

Xossa	$y=\sin x$	$y=\cos x$
XossaAniqlanish sohasi	$y=\sin x$ barcha $x$	$y=\cos x$ barcha $x$
XossaQiymatlar sohasi	$y=\sin x$ $[-1;\,1]$	$y=\cos x$ $[-1;\,1]$
XossaJuft-toqligi	$y=\sin x$ toq	$y=\cos x$ juft
XossaEng kichik musbat davri	$y=\sin x$ $2\pi$	$y=\cos x$ $2\pi$
XossaNollari	$y=\sin x$ $x=\pi n$	$y=\cos x$ $x=\dfrac{\pi}{2}+\pi n$
XossaEng katta qiymat	$y=\sin x$ $1$ , $x=\dfrac{\pi}{2}+2\pi n$ da	$y=\cos x$ $1$ , $x=2\pi n$ da
XossaEng kichik qiymat	$y=\sin x$ $-1$ , $x=-\dfrac{\pi}{2}+2\pi n$ da	$y=\cos x$ $-1$ , $x=\pi+2\pi n$ da

Xossa	$y=\operatorname{tg}x$	$y=\operatorname{ctg}x$
XossaAniqlanish sohasi	$y=\operatorname{tg}x$ $x\ne\dfrac{\pi}{2}+\pi n$	$y=\operatorname{ctg}x$ $x\ne\pi n$
XossaQiymatlar sohasi	$y=\operatorname{tg}x$ barcha sonlar	$y=\operatorname{ctg}x$ barcha sonlar
XossaJuft-toqligi	$y=\operatorname{tg}x$ toq	$y=\operatorname{ctg}x$ toq
XossaEng kichik musbat davri	$y=\operatorname{tg}x$ $\pi$	$y=\operatorname{ctg}x$ $\pi$
XossaNollari	$y=\operatorname{tg}x$ $x=\pi n$	$y=\operatorname{ctg}x$ $x=\dfrac{\pi}{2}+\pi n$
XossaEng katta qiymat	$y=\operatorname{tg}x$ yo'q	$y=\operatorname{ctg}x$ yo'q
XossaEng kichik qiymat	$y=\operatorname{tg}x$ yo'q	$y=\operatorname{ctg}x$ yo'q

Funksiya	O'sish oraliqlari	Kamayish oraliqlari
Funksiya $\sin x$	O'sish oraliqlari $\left[-\dfrac{\pi}{2}+2\pi n;\,\dfrac{\pi}{2}+2\pi n\right]$	Kamayish oraliqlari $\left[\dfrac{\pi}{2}+2\pi n;\,\dfrac{3\pi}{2}+2\pi n\right]$
Funksiya $\cos x$	O'sish oraliqlari $[-\pi+2\pi n;\,2\pi n]$	Kamayish oraliqlari $[2\pi n;\,\pi+2\pi n]$
Funksiya $\operatorname{tg}x$	O'sish oraliqlarihar bir $\left(-\dfrac{\pi}{2}+\pi n;\,\dfrac{\pi}{2}+\pi n\right)$ da	Kamayish oraliqlariyo'q
Funksiya $\operatorname{ctg}x$	O'sish oraliqlariyo'q	Kamayish oraliqlarihar bir $(\pi n;\,\pi+\pi n)$ da

Tangens har bir oraliqda o'sadi, lekin butun aniqlanish sohasida o'suvchi emas:

\frac{\pi}{4}<\frac{3\pi}{4}

\operatorname{tg}\frac{\pi}{4}=1>\operatorname{tg}\frac{3\pi}{4}=-1

Taqqoslash shu xossaga tayanadi: $\sin 1$ va $\sin 1{,}5$ ikkalasi o'sish oralig'ida, demak $\sin 1<\sin 1{,}5$ .

y=\sin x

: davri

2\pi

, grafigi koordinata boshiga nisbatan simmetrik

y=\cos x

: davri

2\pi

, grafigi

Oy

o'qiga nisbatan simmetrik

y=\operatorname{tg}x

: davri

\pi

, punktir chiziqlarda funksiya mavjud emas

Davrni topish

$T\ne 0$ soni funksiyaning davri, agar aniqlanish sohasidagi har bir $x$ uchun $x+T$ va $x-T$ ham aniqlanish sohasida bo'lib, $f(x+T)=f(x)$ bo'lsa. Odatda eng kichik musbat davr so'raladi.

T=\frac{2\pi}{|k|}

\sin(kx+b)

\cos(kx+b)

uchun

T=\frac{\pi}{|k|}

\operatorname{tg}(kx+b)

\operatorname{ctg}(kx+b)

uchun

Bunda $k\ne 0$ . Oldidagi ko'paytuvchi, qo'shiluvchi va $b$ davrga ta'sir qilmaydi.

Funksiya	Davri	Sababi
Funksiya $y=5\sin\left(3x+\dfrac{\pi}{4}\right)-2$	Davri $\dfrac{2\pi}{3}$	Sababi $k=3$
Funksiya $y=\cos\dfrac{\pi x}{3}$	Davri $6$	Sababi $k=\dfrac{\pi}{3}$ , $\dfrac{2\pi}{k}=6$
Funksiya $y=\operatorname{tg}\dfrac{x}{2}$	Davri $2\pi$	Sababi $k=\dfrac{1}{2}$
Funksiya $y=\sin^2x$	Davri $\pi$	Sababi $\sin^2x=\dfrac{1-\cos 2x}{2}$
Funksiya $y=\cos^2 3x$	Davri $\dfrac{\pi}{3}$	Sababi $\cos^2 3x=\dfrac{1+\cos 6x}{2}$
Funksiya $y=\sin x\cos x$	Davri $\pi$	Sababi $\dfrac{1}{2}\sin 2x$
Funksiya $y=\|\sin x\|$	Davri $\pi$	Sababimanfiy bo'lak yuqoriga aks etadi
Funksiya $y=\|\sin x\|+\|\cos x\|$	Davri $\dfrac{\pi}{2}$	Sababikvadrati $1+\|\sin 2x\|$
Funksiya $y=\sin^4x+\cos^4x$	Davri $\dfrac{\pi}{2}$	Sababi $\dfrac{3+\cos 4x}{4}$
Funksiya $y=\sin^6 3x+\cos^6 3x$	Davri $\dfrac{\pi}{6}$	Sababi $\dfrac{5+3\cos 12x}{8}$

Yig'indi yoki ko'paytmada davrlarning eng kichik umumiy karralisini olamiz. Davrlar $\pi$ ga karrali kasrlar bo'lsa, ko'paytuvchilar uchun quyidagi qoida ishlaydi:

\text{EKUK}\left(\frac{a}{b};\,\frac{c}{d}\right)=\frac{\text{EKUK}(a;\,c)}{\text{EKUB}(b;\,d)}

Bunda kasrlar qisqarmas

Masalan:

y=\sin\frac{3x}{2}+\cos\frac{2x}{5}

20\pi=15\cdot\frac{4\pi}{3}=4\cdot 5\pi

Davrlar

\dfrac{4\pi}{3}

5\pi

, EKUK

20\pi

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