3.1.16
Jab aap do sine ya cosine waves ko add karte ho jo alag-alag frequencies par hain, to result ek aisi single wave jaisi dikhti hai jiska amplitude dheere-dheere hilta rehta hai . Sum-to-product formulas ek sum/difference of trig functions ko product of trig functions mein badal dete hain. Yahi woh cheez hai jo aapka kaan beats ke roop mein sunata hai, aur yahi woh cheez hai jo sin 3 x + sin x = 0
KYA: sin A ± sin B cos A ± cos B KYUN: Products ko zero set karna aasaan hota hai (equations solve karna), aur yeh ek slow "envelope" × KAISE: Product-to-sum identities ko ulta chalao ek clever substitution use karke.
Hum angle addition formulas se shuru karte hain, jinhe hum known maante hain:
sin ( x + y ) = sin x cos y + cos x sin y sin ( x − y ) = sin x cos y − cos x sin y
Inhe Add karo — cos x sin y
\sin(x+y) + \sin(x-y) = 2\sin x\cos y \tag{1}
Inhe Subtract karo — sin x cos y
\sin(x+y) - \sin(x-y) = 2\cos x\sin y \tag{2}
Ab cosine ke liye bhi yahi karo:
cos ( x + y ) = cos x cos y − sin x sin y cos ( x − y ) = cos x cos y + sin x sin y
Add karo: cos ( x + y ) + cos ( x − y ) = 2 cos x cos y
Subtract karo: cos ( x − y ) − cos ( x + y ) = 2 sin x sin y
Intuition Key substitution — KYUN yeh kaam karta haiEquations (1)–(4) mein abhi bhi x + y x − y rename karte hain taaki left side ek simple sum of do angles ban jaaye . Maano ki
A = x + y , B = x − y . ⇒ x = 2 A + B ⇒ y = 2 A − B A , B 2 A + B 2 A − B
Substitution ko (1)–(4) par apply karte hain:
Definition Pattern padhna
Har RHS mein == 2 ==
Dono arguments hamesha half-sum 2 A + B half-difference 2 A − B
Sine sum → sin ⋅ cos sine difference → cos ⋅ sin Cosine "same type" rehta hai: sum → cos cos → − sin sin
sin 3 x + sin x = 0 A = 3 x B = x
2 A + B = 2 4 x = 2 x Kyun? do angles ka half-sum.2 A − B = 2 2 x = x Kyun? half-difference.
sin 3 x + sin x = 2 sin 2 x cos x = 0 Yeh kyun help karta hai: ek product = 0 ka matlab hai ek factor 0 hai. Toh sin 2 x = 0 cos x = 0 sin 2 x = 0 ⇒ 2 x = nπ ⇒ x = 2 nπ cos x = 0 ⇒ x = 2 π + nπ Answer: x = 2 nπ , n ∈ Z
cos 7 5 ∘ + cos 1 5 ∘ A = 7 5 ∘ , B = 1 5 ∘ = 4 5 ∘ = 3 0 ∘ cos 7 5 ∘ + cos 1 5 ∘ = 2 cos 4 5 ∘ cos 3 0 ∘ = 2 ⋅ 2 2 ⋅ 2 3 = 2 6 Yeh step kyun: humne cosine-sum → cos cos 4 5 ∘ , 3 0 ∘
Worked example 3) Beats —sin ( 2 π f 1 t ) + sin ( 2 π f 2 t ) A = 2 π f 1 t B = 2 π f 2 t sin A + sin B = 2 slow envelope cos ( 2 π 2 f 1 − f 2 t ) fast carrier sin ( 2 π 2 f 1 + f 2 t ) Kyun matter karta hai: kaan loudness ko beat frequency ∣ f 1 − f 2 ∣ physics ko explain karta hai .
Worked example 4) Prove karocos A + cos B sin A + sin B = tan 2 A + B 2 sin 2 A + B cos 2 A − B 2 cos 2 A + B cos 2 A − B 2 cos 2 A − B cancel ho jaate hain, aur bacha rehta hai c o s 2 A + B s i n 2 A + B = tan 2 A + B Kyun: factoring ne ek aisa common factor expose kiya jo original form mein invisible tha.
sin A + sin B = sin ( A + B ) Kyun sahi lagta hai: bahar addition aisa lagta hai jaise andar slide ho jaana chahiye, aur logs/exponents sums ke saath acche se behave karte hain. Kyun galat hai: sin sin Fix: sin A + sin B = 2 sin 2 A + B cos 2 A − B A = B = 9 0 ∘ = 2 = 2 sin 9 0 ∘ cos 0 ∘ = 2 sin 18 0 ∘ = 0
cos A − cos B Kyun sahi lagta hai: baaki teen formulas mein clean + 2 Fix: cos A − cos B = − 2 sin 2 A + B sin 2 A − B decreasing hai, toh cosines ka difference sign flip karta hai — minus ka source yahi hai.
Common mistake Sine-difference formula mein sin/cos ko andar swap kar lenaFix: sin A − sin B outer factor cos 2 A + B inner sin 2 A − B sin A + sin B swap ho jaate hain. A = 9 0 ∘ , B = 3 0 ∘
Recall Reveal karne se pehle try karo
cos A + cos B Kaunsa substitution (1)–(4) ko product form mein convert karta hai?
Kaunse formula mein minus sign hota hai, aur physically kyun?
cos 5 x − cos x
Answers: 1. cos ( x + y ) + cos ( x − y ) = 2 cos x cos y x = 2 A + B , y = 2 A − B A = x + y , B = x − y cos A − cos B = − 2 sin 2 A + B sin 2 A − B − 2 sin 3 x sin 2 x
Recall Feynman: ek 12-saal ke bachche ko samjhaoImagine karo do dost almost ek hi speed par taaliyan baja rahe hain. Kabhi-kabhi unki taaliyan ek saath milti hain aur LOUD sunti hain; thodi der baad woh opposite ho jaate hain aur soft sunti hain. Toh do steady taali bajane wale milkar ek slow "loud–soft–loud" pulse banate hain. Sum-to-product woh math hai jo kehti hai: do simple wiggles jab add hoti hain = ek fast wiggle jiske loudness dheere-dheere saanste andar aur bahar. Aur jab total bilkul zero ho, toh matlab sirf ek dono pieces zero hai — isliye ek sum ko multiplication mein badalna solving ko itna aasaan bana deta hai.
Mnemonic Argument ke + signs yaad rakho"Half the SUM, half the DIFFERENCE, times 2."
Signs: S in-sum → S in× C os; S in-diff swap karke C os× S in ho jaata hai. C os-sum → C os× C os. C os-diff odd one out hai: − 2 sin sin D ifference ko D own/minus milta hai).
sin A + sin B 2 sin 2 A + B cos 2 A − B
sin A − sin B 2 cos 2 A + B sin 2 A − B
cos A + cos B 2 cos 2 A + B cos 2 A − B
cos A − cos B − 2 sin 2 A + B sin 2 A − B
Kaunse sum-to-product formula mein minus sign hota hai? cos A − cos B
Kaunsa substitution inhe angle-addition se derive karta hai? A = x + y , B = x − y ⇒ x = 2 A + B , y = 2 A − B
c o s A + c o s B s i n A + s i n B tan 2 A + B
sin 3 x + sin x 2 sin 2 x cos x
Beats mein, sin ( 2 π f 1 t ) + sin ( 2 π f 2 t ) 2 ∣ f 1 − f 2 ∣ ∣ f 1 − f 2 ∣
cos 7 5 ∘ + cos 1 5 ∘ sin sum-diff -> 2 sinx cosy
cos sum-diff -> 2 cosx cosy etc
Substitution A=x+y, B=x-y
Four sum-to-product identities
Pattern: factor 2, half-sum, half-diff
sine sum -> sin cos; diff swaps
cosine diff has minus sign
Beats: envelope x carrier
