Ushbu darsda ikki burchak yig‘indisi va ayirmasining trigonometrik funksiyalarini hisoblash imkonini beradigan qo‘shish formulalari haqida so‘z yuritamiz. Bu formulalar butun trigonometriyaning "poydevori" — undan keyingi barcha formulalar shulardan kelib chiqadi.
§1. Sinus va kosinus uchun qo‘shish formulalari
Ta’rif. Ikki burchak yig‘indisi yoki ayirmasining (α ± β \alpha\pm\beta α ± β qo‘shish formulalari deyiladi.
α
β
Qoida. Kosinus uchun:
cos ( α + β ) = cos α cos β − sin α sin β , \cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta,
cos ( α + β ) = cos α cos β − sin α sin β , cos ( α − β ) = cos α cos β + sin α sin β . \cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta.
cos ( α − β ) = cos α cos β + sin α sin β .
Qoida. Sinus uchun:
sin ( α + β ) = sin α cos β + cos α sin β , \sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta,
sin ( α + β ) = sin α cos β + cos α sin β , sin ( α − β ) = sin α cos β − cos α sin β . \sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta.
sin ( α − β ) = sin α cos β − cos α sin β .
Eslatma. Ishoraga diqqat: kosinusda ishora "teskari" bo‘ladi (yig‘indida minus, ayirmada plyus), sinusda esa "bir xil" (yig‘indida plyus, ayirmada minus). Bu chalkashmaslikning kaliti.
Savol. Nega bu formulalar kerak? Chunki cos 75 ∘ \cos75^\circ cos 7 5 ∘ 75 ∘ = 45 ∘ + 30 ∘ 75^\circ=45^\circ+30^\circ 7 5 ∘ = 4 5 ∘ + 3 0 ∘
Misol. cos 75 ∘ \cos75^\circ cos 7 5 ∘ Yechish. 75 ∘ = 45 ∘ + 30 ∘ 75^\circ=45^\circ+30^\circ 7 5 ∘ = 4 5 ∘ + 3 0 ∘
cos 75 ∘ = cos 45 ∘ cos 30 ∘ − sin 45 ∘ sin 30 ∘ . \cos75^\circ=\cos45^\circ\cos30^\circ-\sin45^\circ\sin30^\circ.
cos 7 5 ∘ = cos 4 5 ∘ cos 3 0 ∘ − sin 4 5 ∘ sin 3 0 ∘ . Jadval qiymatlarini qo‘yamiz:
= 2 2 ⋅ 3 2 − 2 2 ⋅ 1 2 = 6 4 − 2 4 = 6 − 2 4 . =\frac{\sqrt2}{2}\cdot\frac{\sqrt3}{2}-\frac{\sqrt2}{2}\cdot\frac{1}{2}=\frac{\sqrt6}{4}-\frac{\sqrt2}{4}=\frac{\sqrt6-\sqrt2}{4}.
= 2 2 ⋅ 2 3 − 2 2 ⋅ 2 1 = 4 6 − 4 2 = 4 6 − 2 . Javob. 6 − 2 4 \dfrac{\sqrt6-\sqrt2}{4} 4 6 − 2
Misol. sin ( α + β ) \sin(\alpha+\beta) sin ( α + β ) sin α = 3 5 \sin\alpha=\frac{3}{5} sin α = 5 3 α \alpha α cos β = 5 13 \cos\beta=\frac{5}{13} cos β = 13 5 β \beta β Yechish. Avval yetishmagan qiymatlarni topamiz: cos α = 4 5 \cos\alpha=\frac{4}{5} cos α = 5 4 sin β = 12 13 \sin\beta=\frac{12}{13} sin β = 13 12
sin ( α + β ) = 3 5 ⋅ 5 13 + 4 5 ⋅ 12 13 = 15 65 + 48 65 = 63 65 . \sin(\alpha+\beta)=\frac{3}{5}\cdot\frac{5}{13}+\frac{4}{5}\cdot\frac{12}{13}=\frac{15}{65}+\frac{48}{65}=\frac{63}{65}.
sin ( α + β ) = 5 3 ⋅ 13 5 + 5 4 ⋅ 13 12 = 65 15 + 65 48 = 65 63 . Javob. 63 65 \dfrac{63}{65} 65 63
§2. Tangens uchun qo‘shish formulalari
Qoida. $$ \tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta},\qquad \tan(\alpha-\beta)=\frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}. $$
Misol. tan 75 ∘ \tan75^\circ tan 7 5 ∘ Yechish. 75 ∘ = 45 ∘ + 30 ∘ 75^\circ=45^\circ+30^\circ 7 5 ∘ = 4 5 ∘ + 3 0 ∘ tan 45 ∘ = 1 \tan45^\circ=1 tan 4 5 ∘ = 1 tan 30 ∘ = 1 3 \tan30^\circ=\frac{1}{\sqrt3} tan 3 0 ∘ = 3 1
tan 75 ∘ = 1 + 1 3 1 − 1 ⋅ 1 3 = 3 + 1 3 − 1 . \tan75^\circ=\frac{1+\frac{1}{\sqrt3}}{1-1\cdot\frac{1}{\sqrt3}}=\frac{\sqrt3+1}{\sqrt3-1}.
tan 7 5 ∘ = 1 − 1 ⋅ 3 1 1 + 3 1 = 3 − 1 3 + 1 . Maxrajni ratsionallashtiramiz (surat va maxrajni 3 + 1 \sqrt3+1 3 + 1
= ( 3 + 1 ) 2 ( 3 ) 2 − 1 2 = 3 + 2 3 + 1 2 = 4 + 2 3 2 = 2 + 3 . =\frac{(\sqrt3+1)^2}{(\sqrt3)^2-1^2}=\frac{3+2\sqrt3+1}{2}=\frac{4+2\sqrt3}{2}=2+\sqrt3.
= ( 3 ) 2 − 1 2 ( 3 + 1 ) 2 = 2 3 + 2 3 + 1 = 2 4 + 2 3 = 2 + 3 . Javob. 2 + 3 2+\sqrt3 2 + 3
§3. Formulalarni ifoda soddalashtirishda qo‘llash
Qo‘shish formulalarini ko‘pincha teskari yo‘nalishda — yig‘indini bitta funksiyaga yig‘ish uchun ishlatamiz.
Misol. sin 40 ∘ cos 20 ∘ + cos 40 ∘ sin 20 ∘ \sin40^\circ\cos20^\circ+\cos40^\circ\sin20^\circ sin 4 0 ∘ cos 2 0 ∘ + cos 4 0 ∘ sin 2 0 ∘ Yechish. Bu aynan sin ( α + β ) \sin(\alpha+\beta) sin ( α + β ) α = 40 ∘ \alpha=40^\circ α = 4 0 ∘ β = 20 ∘ \beta=20^\circ β = 2 0 ∘
sin 40 ∘ cos 20 ∘ + cos 40 ∘ sin 20 ∘ = sin ( 40 ∘ + 20 ∘ ) = sin 60 ∘ = 3 2 . \sin40^\circ\cos20^\circ+\cos40^\circ\sin20^\circ=\sin(40^\circ+20^\circ)=\sin60^\circ=\frac{\sqrt3}{2}.
sin 4 0 ∘ cos 2 0 ∘ + cos 4 0 ∘ sin 2 0 ∘ = sin ( 4 0 ∘ + 2 0 ∘ ) = sin 6 0 ∘ = 2 3 . Javob. 3 2 \dfrac{\sqrt3}{2} 2 3
Misol. cos ( α + β ) + cos ( α − β ) \cos(\alpha+\beta)+\cos(\alpha-\beta) cos ( α + β ) + cos ( α − β ) Yechish. Ikkala formulani yozib qo‘shamiz:
( cos α cos β − sin α sin β ) + ( cos α cos β + sin α sin β ) = 2 cos α cos β . (\cos\alpha\cos\beta-\sin\alpha\sin\beta)+(\cos\alpha\cos\beta+\sin\alpha\sin\beta)=2\cos\alpha\cos\beta.
( cos α cos β − sin α sin β ) + ( cos α cos β + sin α sin β ) = 2 cos α cos β . Javob. 2 cos α cos β 2\cos\alpha\cos\beta 2 cos α cos β
§ Lug‘at
qo‘shish formulalari — addition formulas