Kirish

Ikkilangan va yarim burchak formulalari

sin⁡α=35\sin\alpha=\dfrac{3}{5} o'tkir burchak uchun sin⁡2α\sin 2\alpha ikki baravar katta emas: u 65\dfrac{6}{5} emas, 2425\dfrac{24}{25}. Qo'shish formulalarida β=α\beta=\alpha deb olsak, karrali burchak formulalari chiqadi. Ular darajani pasaytirish, ko'paytmani yig'ish va yarim burchakni topishga xizmat qiladi.

Ikkilangan burchak

sin⁡2α=2sin⁡αcos⁡α\sin 2\alpha=2\sin\alpha\cos\alpha
Har qanday α\alpha uchun
cos⁡2α=cos⁡2α−sin⁡2α\cos 2\alpha=\cos^2\alpha-\sin^2\alpha
=2cos⁡2α−1=2\cos^2\alpha-1
=1−2sin⁡2α=1-2\sin^2\alpha
Uch ko'rinish, berilganiga qarab tanlanadi
tg⁡2α=2tg⁡α1−tg⁡2α\operatorname{tg}2\alpha=\frac{2\operatorname{tg}\alpha}{1-\operatorname{tg}^2\alpha}
ctg⁡2α=ctg⁡2α−12ctg⁡α\operatorname{ctg}2\alpha=\frac{\operatorname{ctg}^2\alpha-1}{2\operatorname{ctg}\alpha}
Tangens: cos⁡α≠0\cos\alpha\ne 0, cos⁡2α≠0\cos 2\alpha\ne 0. Kotangens: sin⁡2α≠0\sin 2\alpha\ne 0

tg⁡α\operatorname{tg}\alpha orqali sinus va kosinus (universal almashtirish). Bu formulalar cos⁡α≠0\cos\alpha\ne 0 da o'rinli va hech qanday chorak so'ramaydi:

sin⁡2α=2tg⁡α1+tg⁡2α\sin 2\alpha=\frac{2\operatorname{tg}\alpha}{1+\operatorname{tg}^2\alpha}
cos⁡2α=1−tg⁡2α1+tg⁡2α\cos 2\alpha=\frac{1-\operatorname{tg}^2\alpha}{1+\operatorname{tg}^2\alpha}

Masalan, tg⁡α=3\operatorname{tg}\alpha=3 bo'lsa:

sin⁡2α=610=35\sin 2\alpha=\frac{6}{10}=\frac{3}{5}
cos⁡2α=−810=−45\cos 2\alpha=\frac{-8}{10}=-\frac{4}{5}
tg⁡2α=6−8=−34\operatorname{tg}2\alpha=\frac{6}{-8}=-\frac{3}{4}

Yozuvga e'tibor bering: ba'zi masalalarda cos⁡−12α\cos^{-1}2\alpha deb 1cos⁡2α\dfrac{1}{\cos 2\alpha} yoziladi. Masalan, tg⁡α=3\operatorname{tg}\alpha=3 da cos⁡−12α=−54\cos^{-1}2\alpha=-\dfrac{5}{4}. Bu yozuv ikki ma'noli: kalkulyatorlarda va ko'p xorijiy kitoblarda cos⁡−1\cos^{-1} arkkosinusni bildiradi. Bizda arkkosinus doim arccos⁡\arccos deb yoziladi, teskari sonni esa aniq qilib (cos⁡2α)−1(\cos 2\alpha)^{-1} deb yozish mumkin.

Formulaning qo'shimcha ko'rinishlari

TenglikSharti
1+sin⁡2α=(sin⁡α+cos⁡α)21+\sin 2\alpha=(\sin\alpha+\cos\alpha)^2har qanday α\alpha
1−sin⁡2α=(sin⁡α−cos⁡α)21-\sin 2\alpha=(\sin\alpha-\cos\alpha)^2har qanday α\alpha
1+cos⁡2α=2cos⁡2α1+\cos 2\alpha=2\cos^2\alphahar qanday α\alpha
1−cos⁡2α=2sin⁡2α1-\cos 2\alpha=2\sin^2\alphahar qanday α\alpha
cos⁡4α−sin⁡4α=cos⁡2α\cos^4\alpha-\sin^4\alpha=\cos 2\alphahar qanday α\alpha
tg⁡α+ctg⁡α=2sin⁡2α\operatorname{tg}\alpha+\operatorname{ctg}\alpha=\dfrac{2}{\sin 2\alpha}sin⁡2α≠0\sin 2\alpha\ne 0
ctg⁡α−tg⁡α=2ctg⁡2α\operatorname{ctg}\alpha-\operatorname{tg}\alpha=2\operatorname{ctg}2\alphasin⁡2α≠0\sin 2\alpha\ne 0
sin⁡4α+cos⁡4α=1−12sin⁡22α\sin^4\alpha+\cos^4\alpha=1-\frac{1}{2}\sin^2 2\alpha
=3+cos⁡4α4=\frac{3+\cos 4\alpha}{4}
Har qanday α\alpha uchun
sin⁡6α+cos⁡6α=1−34sin⁡22α\sin^6\alpha+\cos^6\alpha=1-\frac{3}{4}\sin^2 2\alpha
=5+3cos⁡4α8=\frac{5+3\cos 4\alpha}{8}
Har qanday α\alpha uchun

Ildiz ostida bunday ifoda kelsa, modul paydo bo'ladi: 1+cos⁡2α=2 ∣cos⁡α∣\sqrt{1+\cos 2\alpha}=\sqrt{2}\,|\cos\alpha|, 1+sin⁡2α=∣sin⁡α+cos⁡α∣\sqrt{1+\sin 2\alpha}=|\sin\alpha+\cos\alpha|.

Darajani pasaytirish

sin⁡2α=1−cos⁡2α2\sin^2\alpha=\frac{1-\cos 2\alpha}{2}
cos⁡2α=1+cos⁡2α2\cos^2\alpha=\frac{1+\cos 2\alpha}{2}
Har qanday α\alpha uchun. Kvadrat yo'qoladi, burchak ikki baravar kattalashadi
sin⁡3α=3sin⁡α−sin⁡3α4\sin^3\alpha=\frac{3\sin\alpha-\sin 3\alpha}{4}
cos⁡3α=3cos⁡α+cos⁡3α4\cos^3\alpha=\frac{3\cos\alpha+\cos 3\alpha}{4}
Uchlangan burchak formulalaridan

Yarim burchak

Darajani pasaytirish formulasida α\alpha o'rniga α2\dfrac{\alpha}{2} qo'yamiz.

sin⁡2α2=1−cos⁡α2\sin^2\frac{\alpha}{2}=\frac{1-\cos\alpha}{2}
cos⁡2α2=1+cos⁡α2\cos^2\frac{\alpha}{2}=\frac{1+\cos\alpha}{2}
Har qanday α\alpha uchun
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