Kirish

Yig'indini ko'paytmaga va ko'paytmani yig'indiga keltirish

sin⁡75∘+sin⁡15∘\sin 75^\circ+\sin 15^\circ yig'indining har bir hadi jadvalda yo'q, lekin ularning yarim yig'indisi 45∘45^\circ, yarim ayirmasi 30∘30^\circ, shuning uchun javob toza chiqadi: 62\dfrac{\sqrt{6}}{2}. Yig'indini ko'paytmaga aylantirish tenglamalarni ko'paytuvchilarga ajratadi, ko'paytmani yig'indiga aylantirish esa murakkab ko'paytmalarni hisoblashga imkon beradi.

Ko'paytmani yig'indiga keltirish

Qo'shish formulalarini qo'shamiz yoki ayiramiz. Masalan:

sin⁡(α+β)+sin⁡(α−β)\sin(\alpha+\beta)+\sin(\alpha-\beta)
=2sin⁡αcos⁡β=2\sin\alpha\cos\beta

Shu yo'l bilan uchta formula chiqadi:

sin⁡αcos⁡β\sin\alpha\cos\beta
=12(sin⁡(α+β)+sin⁡(α−β))=\frac{1}{2}\left(\sin(\alpha+\beta)+\sin(\alpha-\beta)\right)
Har qanday burchaklar uchun
cos⁡αcos⁡β\cos\alpha\cos\beta
=12(cos⁡(α−β)+cos⁡(α+β))=\frac{1}{2}\left(\cos(\alpha-\beta)+\cos(\alpha+\beta)\right)
Har qanday burchaklar uchun
sin⁡αsin⁡β\sin\alpha\sin\beta
=12(cos⁡(α−β)−cos⁡(α+β))=\frac{1}{2}\left(\cos(\alpha-\beta)-\cos(\alpha+\beta)\right)
Ayirma birinchi turadi, yig'indi ikkinchi

Ikkinchi burchak manfiy chiqsa, sinus minus beradi, kosinus bermaydi:

sin⁡18∘cos⁡72∘\sin 18^\circ\cos 72^\circ
=12(sin⁡90∘+sin⁡(−54∘))=\frac{1}{2}(\sin 90^\circ+\sin(-54^\circ))
=12(1−sin⁡54∘)=\frac{1}{2}(1-\sin 54^\circ)

Yig'indini ko'paytmaga keltirish

Yuqoridagi formulalarni teskari o'qiymiz: α+β=x\alpha+\beta=x, α−β=y\alpha-\beta=y deb belgilasak, α=x+y2\alpha=\dfrac{x+y}{2}, β=x−y2\beta=\dfrac{x-y}{2}.

Yig'indiKo'paytma
sin⁡α+sin⁡β\sin\alpha+\sin\beta2sin⁡α+β2cos⁡α−β22\sin\dfrac{\alpha+\beta}{2}\cos\dfrac{\alpha-\beta}{2}
sin⁡α−sin⁡β\sin\alpha-\sin\beta2sin⁡α−β2cos⁡α+β22\sin\dfrac{\alpha-\beta}{2}\cos\dfrac{\alpha+\beta}{2}
cos⁡α+cos⁡β\cos\alpha+\cos\beta2cos⁡α+β2cos⁡α−β22\cos\dfrac{\alpha+\beta}{2}\cos\dfrac{\alpha-\beta}{2}
cos⁡α−cos⁡β\cos\alpha-\cos\beta−2sin⁡α+β2sin⁡α−β2-2\sin\dfrac{\alpha+\beta}{2}\sin\dfrac{\alpha-\beta}{2}

Bu to'rt formula har qanday burchaklar uchun o'rinli. Kosinuslar ayirmasidagi minusni unutmang. Uni boshqacha ham yozish mumkin:

cos⁡α−cos⁡β\cos\alpha-\cos\beta
=2sin⁡α+β2sin⁡β−α2=2\sin\frac{\alpha+\beta}{2}\sin\frac{\beta-\alpha}{2}

Tangens va kotangens uchun ham shunday formulalar bor. Ular bir maxrajga keltirish va qo'shish formulasidan chiqadi.

Yig'indiKo'paytma ko'rinishiSharti
Yig'inditg⁡α+tg⁡β\operatorname{tg}\alpha+\operatorname{tg}\betaKo'paytma ko'rinishisin⁡(α+β)cos⁡αcos⁡β\dfrac{\sin(\alpha+\beta)}{\cos\alpha\cos\beta}Sharticos⁡αcos⁡β≠0\cos\alpha\cos\beta\ne 0
Yig'inditg⁡α−tg⁡β\operatorname{tg}\alpha-\operatorname{tg}\betaKo'paytma ko'rinishisin⁡(α−β)cos⁡αcos⁡β\dfrac{\sin(\alpha-\beta)}{\cos\alpha\cos\beta}Sharticos⁡αcos⁡β≠0\cos\alpha\cos\beta\ne 0
Yig'indictg⁡α+ctg⁡β\operatorname{ctg}\alpha+\operatorname{ctg}\betaKo'paytma ko'rinishisin⁡(α+β)sin⁡αsin⁡β\dfrac{\sin(\alpha+\beta)}{\sin\alpha\sin\beta}Shartisin⁡αsin⁡β≠0\sin\alpha\sin\beta\ne 0
Yig'indictg⁡α−ctg⁡β\operatorname{ctg}\alpha-\operatorname{ctg}\betaKo'paytma ko'rinishisin⁡(β−α)sin⁡αsin⁡β\dfrac{\sin(\beta-\alpha)}{\sin\alpha\sin\beta}Shartisin⁡αsin⁡β≠0\sin\alpha\sin\beta\ne 0
Yig'inditg⁡α+ctg⁡β\operatorname{tg}\alpha+\operatorname{ctg}\betaKo'paytma ko'rinishicos⁡(α−β)cos⁡αsin⁡β\dfrac{\cos(\alpha-\beta)}{\cos\alpha\sin\beta}Sharticos⁡αsin⁡β≠0\cos\alpha\sin\beta\ne 0

Sinus va kosinus aralash kelganda

Formulalar bir xil nomdagi funksiyalar uchun. Aralash kelsa, avval keltirish formulasi bilan bir xil nomga o'tamiz:

sin⁡70∘+cos⁡40∘\sin 70^\circ+\cos 40^\circ
=sin⁡70∘+sin⁡50∘=\sin 70^\circ+\sin 50^\circ
=2sin⁡60∘cos⁡10∘=3cos⁡10∘=2\sin 60^\circ\cos 10^\circ=\sqrt{3}\cos 10^\circ
sin⁡α+cos⁡α=2sin⁡(α+π4)\sin\alpha+\cos\alpha=\sqrt{2}\sin\left(\alpha+\frac{\pi}{4}\right)
sin⁡α−cos⁡α=2sin⁡(α−π4)\sin\alpha-\cos\alpha=\sqrt{2}\sin\left(\alpha-\frac{\pi}{4}\right)

Kvadratlar ayirmasi

Kvadratlar ayirmasini ko'paytuvchilarga ajratib, so'ng yuqoridagi formulalarni qo'llasak, juda qisqa natija chiqadi:

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