Kirish

Qo'shish formulalari

sin⁡75∘\sin 75^\circ jadvalda yo'q, lekin 75∘=45∘+30∘75^\circ=45^\circ+30^\circ. Qo'shish formulalari ikki burchak yig'indisi yoki ayirmasining funksiyasini har bir burchakning funksiyalari orqali ifodalaydi: sin⁡75∘=6+24\sin 75^\circ=\dfrac{\sqrt{6}+\sqrt{2}}{4}. Trigonometriyaning qolgan formulalari shulardan chiqadi.

Kosinus formulasining isboti

M(cos⁡α; sin⁡α)M(\cos\alpha;\,\sin\alpha) va N(cos⁡β; sin⁡β)N(\cos\beta;\,\sin\beta) birlik aylanadagi nuqtalar bo'lsin. OM→\overrightarrow{OM} va ON→\overrightarrow{ON} uzunligi 11 ga teng vektorlar, ular orasidagi burchak kosinusi cos⁡(α−β)\cos(\alpha-\beta) ga teng. Skalyar ko'paytma koordinatalar orqali ham hisoblanadi, shuning uchun:

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

β\beta o'rniga −β-\beta qo'ysak, yig'indi kosinusi chiqadi. sin⁡γ=cos⁡(π2−γ)\sin\gamma=\cos\left(\dfrac{\pi}{2}-\gamma\right) dan sinus uchun formula chiqadi. Tangens esa sinusning kosinusga nisbati.

Qo'shish formulalari

FunksiyaYig'indiAyirma
Funksiyasin⁡\sinYig'indisin⁡(α+β)=\sin(\alpha+\beta)= sin⁡αcos⁡β+cos⁡αsin⁡β\sin\alpha\cos\beta+\cos\alpha\sin\betaAyirmasin⁡(α−β)=\sin(\alpha-\beta)= sin⁡αcos⁡β−cos⁡αsin⁡β\sin\alpha\cos\beta-\cos\alpha\sin\beta
Funksiyacos⁡\cosYig'indicos⁡(α+β)=\cos(\alpha+\beta)= cos⁡αcos⁡β−sin⁡αsin⁡β\cos\alpha\cos\beta-\sin\alpha\sin\betaAyirmacos⁡(α−β)=\cos(\alpha-\beta)= cos⁡αcos⁡β+sin⁡αsin⁡β\cos\alpha\cos\beta+\sin\alpha\sin\beta
Funksiyatg⁡\operatorname{tg}Yig'inditg⁡(α+β)=tg⁡α+tg⁡β1−tg⁡αtg⁡β\operatorname{tg}(\alpha+\beta)=\dfrac{\operatorname{tg}\alpha+\operatorname{tg}\beta}{1-\operatorname{tg}\alpha\operatorname{tg}\beta}Ayirmatg⁡(α−β)=tg⁡α−tg⁡β1+tg⁡αtg⁡β\operatorname{tg}(\alpha-\beta)=\dfrac{\operatorname{tg}\alpha-\operatorname{tg}\beta}{1+\operatorname{tg}\alpha\operatorname{tg}\beta}
Funksiyactg⁡\operatorname{ctg}Yig'indictg⁡(α+β)=ctg⁡αctg⁡β−1ctg⁡α+ctg⁡β\operatorname{ctg}(\alpha+\beta)=\dfrac{\operatorname{ctg}\alpha\operatorname{ctg}\beta-1}{\operatorname{ctg}\alpha+\operatorname{ctg}\beta}Ayirmactg⁡(α−β)=ctg⁡αctg⁡β+1ctg⁡β−ctg⁡α\operatorname{ctg}(\alpha-\beta)=\dfrac{\operatorname{ctg}\alpha\operatorname{ctg}\beta+1}{\operatorname{ctg}\beta-\operatorname{ctg}\alpha}

Sinus va kosinus formulalari har qanday burchaklar uchun o'rinli. Sinusda ishora saqlanadi, kosinusda almashadi. Tangens formulasi uchun tg⁡α, tg⁡β\operatorname{tg}\alpha,\ \operatorname{tg}\beta va chapdagi tangens mavjud bo'lishi kerak: cos⁡α≠0\cos\alpha\ne 0, cos⁡β≠0\cos\beta\ne 0, cos⁡(α±β)≠0\cos(\alpha\pm\beta)\ne 0. Kotangens formulasi uchun esa sin⁡α≠0\sin\alpha\ne 0, sin⁡β≠0\sin\beta\ne 0, sin⁡(α±β)≠0\sin(\alpha\pm\beta)\ne 0.

Tangens formulasining sharti amaliy ahamiyatga ega: tg⁡αtg⁡β=1\operatorname{tg}\alpha\operatorname{tg}\beta=1 bo'lsa, maxraj nolga aylanadi, tg⁡(α+β)\operatorname{tg}(\alpha+\beta) mavjud emas va α+β=π2+πk\alpha+\beta=\dfrac{\pi}{2}+\pi k. Masalan, tg⁡α=2\operatorname{tg}\alpha=2, tg⁡β=12\operatorname{tg}\beta=\dfrac{1}{2} bo'lgan o'tkir burchaklarda α+β=π2\alpha+\beta=\dfrac{\pi}{2}.

Jadvalda yo'q burchaklar

15∘=45∘−30∘15^\circ=45^\circ-30^\circ, 75∘=45∘+30∘75^\circ=45^\circ+30^\circ, 105∘=60∘+45∘105^\circ=60^\circ+45^\circ bo'lgani uchun quyidagi qiymatlar chiqadi. Ularni keyingi darslarda ham ishlatamiz.

α\alphasin⁡α\sin\alphacos⁡α\cos\alphatg⁡α\operatorname{tg}\alphactg⁡α\operatorname{ctg}\alpha
α\alpha15∘15^\circsin⁡α\sin\alpha6−24\dfrac{\sqrt{6}-\sqrt{2}}{4}cos⁡α\cos\alpha6+24\dfrac{\sqrt{6}+\sqrt{2}}{4}tg⁡α\operatorname{tg}\alpha2−32-\sqrt{3}ctg⁡α\operatorname{ctg}\alpha2+32+\sqrt{3}
α\alpha75∘75^\circsin⁡α\sin\alpha6+24\dfrac{\sqrt{6}+\sqrt{2}}{4}cos⁡α\cos\alpha6−24\dfrac{\sqrt{6}-\sqrt{2}}{4}tg⁡α\operatorname{tg}\alpha2+32+\sqrt{3}ctg⁡α\operatorname{ctg}\alpha2−32-\sqrt{3}
α\alpha105∘105^\circsin⁡α\sin\alpha6+24\dfrac{\sqrt{6}+\sqrt{2}}{4}cos⁡α\cos\alpha2−64\dfrac{\sqrt{2}-\sqrt{6}}{4}tg⁡α\operatorname{tg}\alpha−2−3-2-\sqrt{3}ctg⁡α\operatorname{ctg}\alpha3−2\sqrt{3}-2

Formulani teskari o'qish

Ko'p misolda o'ng tomon berilgan bo'ladi, uni tanib bitta funksiyaga yig'amiz.

IfodaYig'ilganiQiymati
Ifodasin⁡50∘cos⁡10∘+cos⁡50∘sin⁡10∘\sin 50^\circ\cos 10^\circ+\cos 50^\circ\sin 10^\circYig'ilganisin⁡60∘\sin 60^\circQiymati32\dfrac{\sqrt{3}}{2}
Ifodacos⁡70∘cos⁡10∘+sin⁡70∘sin⁡10∘\cos 70^\circ\cos 10^\circ+\sin 70^\circ\sin 10^\circYig'ilganicos⁡60∘\cos 60^\circQiymati12\dfrac{1}{2}
Ifodacos⁡25∘cos⁡20∘−sin⁡25∘sin⁡20∘\cos 25^\circ\cos 20^\circ-\sin 25^\circ\sin 20^\circYig'ilganicos⁡45∘\cos 45^\circQiymati22\dfrac{\sqrt{2}}{2}
Ifodatg⁡25∘+tg⁡20∘1−tg⁡25∘tg⁡20∘\dfrac{\operatorname{tg}25^\circ+\operatorname{tg}20^\circ}{1-\operatorname{tg}25^\circ\operatorname{tg}20^\circ}Yig'ilganitg⁡45∘\operatorname{tg}45^\circQiymati11
1+tg⁡x1−tg⁡x=tg⁡(π4+x)\frac{1+\operatorname{tg}x}{1-\operatorname{tg}x}=\operatorname{tg}\left(\frac{\pi}{4}+x\right)
1−tg⁡x1+tg⁡x=tg⁡(π4−x)\frac{1-\operatorname{tg}x}{1+\operatorname{tg}x}=\operatorname{tg}\left(\frac{\pi}{4}-x\right)
tg⁡π4=1\operatorname{tg}\dfrac{\pi}{4}=1 dan. Shart: tg⁡x\operatorname{tg}x mavjud va maxraj nolga teng emas

Formulalardan kelib chiqadigan tengliklar

Ikki formulani ko'paytirsak:

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