3.1.20
Intuition Bada idea kya haiPythagoras theorem sirf right triangles ke liye kaam karta hai: c 2 = a 2 + b 2 right-angled nahi hote . Law of Cosines essentially Pythagoras hi hai,
plus ek correction term jo angle ke 90°
Agar angle exactly 90°
Agar angle acute ho (< 90° c chota karti hai.
Agar angle obtuse ho (> 90° c bada karti hai.
Yeh KYUN zaroori hai: iske saath tum koi bhi triangle solve kar sakte ho jab tumhe pata ho
(a) do sides + unke beech ka angle (SAS), ya (b) teeno sides (SSS) — yeh woh cases hain
jo Law of Sines akele shuru nahi kar sakti.
Definition Law of CosinesKisi bhi triangle mein jiske sides a , b , c A , B , C c 2 = a 2 + b 2 − 2 ab cos C a 2 = b 2 + c 2 − 2 b c cos A b 2 = a 2 + c 2 − 2 a c cos B correction term hai == − 2 ab cos C ==
Letters ka MATLAB: C a b beech mein hai;
c opposite hai. Angle aur uski opposite side hamesha saath aati hain.
Intuition Correction ka sign "minus" kyun haiAcute angles ke liye cos C > 0 subtract karte hain → c C → 90° cos C → 0 C cos C < 0 − 2 ab cos C > 0 length add karta hai.
Definition ko cos C cos C = 2 ab a 2 + b 2 − c 2 Kyun useful hai? Teeno sides dene par, yeh seedha arccos
Worked example SAS — teesri side dhundhoTriangle jisme a = 8 b = 5 C = 60° c c 2 = 8 2 + 5 2 − 2 ( 8 ) ( 5 ) cos 60° Yeh step kyun? SAS mein C c 2 = 64 + 25 − 80 ( 0.5 ) = 89 − 40 = 49 ⇒ c = 7 cos 60° = 0.5 − 40
Worked example SSS — angle dhundhoSides a = 7 b = 8 c = 9 C 9 cos C = 2 ( 7 ) ( 8 ) 7 2 + 8 2 − 9 2 = 112 49 + 64 − 81 = 112 32 = 0.2857 Yeh arrangement kyun? C c = 9 c a , b C = arccos ( 0.2857 ) ≈ 73.4°
Worked example Obtuse checka = 3 b = 4 C = 120° cos 120° = − 0.5 c 2 = 9 + 16 − 2 ( 3 ) ( 4 ) ( − 0.5 ) = 25 + 12 = 37 ⇒ c ≈ 6.08 90° 5 + 12 add karta hai — correction ka sign flip ho jaata hai.
Recall Compute karne se pehle forecast karoTriangle a = 6 , b = 6 , C = 90° Predict karo c aage padhne se pehle .
Verify: c 2 = 36 + 36 − 2 ( 36 ) ( 0 ) = 72 c = 6 2 ≈ 8.49 90°
Common mistake Galat angle ko galat side ke saath pair karnaGalat jo sahi lagta hai: "Main angle A b c − 2 b c cos B Kyun tempting lagta hai: saare letters idhar-udhar ghoom rahe hote hain aur mismatch karna aasaan hai.
Fix: Correction term mein angle wahi hona chahiye jo left side wali side ke opposite ho. c 2 = ⋯ − 2 ab cos C c C a , b neighbours hain.
Common mistake SAS mein galat angle use karnaGalat: a , b non-included angle, toh use seedha plug in karna.
Fix: Law of Cosines ke SAS ko angle do di hui sides ke beech chahiye. Agar angle unke beech mein nahi hai, toh yeh ambiguous SSA case hai — iske liye Law of Sines use karo.
Common mistake Obtuse angles ke liye sign drop karnaGalat: cos 120° + 0.5 Fix: cos negative hota hai. Sign rakhna hi woh cheez hai jo obtuse triangles ko lamba banata hai.
Recall Ek 12-saal ke bachche ko samjhaoSocho do sticks ek hinge se judi hain. Unke door wale siron ke beech ek gap hai. Agar hinge thoda sa kholo
(chota angle), toh sire paas hain, toh gap chota hai. Zyada kholo (bada angle) aur
sire door ho jaate hain, toh gap bada ho jaata hai. Law of Cosines bas ek formula hai jo tumhe gap ki
length batata hai do stick lengths aur hinge kitna khola hai isse. Jab hinge ek
perfect corner (90°
"SOS — Same, Opposite, Subtract cos."
Left side = akele side ka S quare; right = baaki do ke S quares ka sum; phir
S ubtract karo 2 × × cos O pposite angle).
Yeh bhi: "Jis angle ka cosine lete ho, woh akeli side ke saamne wala hona chahiye."
Side c c 2 = a 2 + b 2 − 2 ab cos C
c 2 = a 2 + b 2 − 2 ab cos C Angle C c a b
Jab C = 90° Pythagoras: c 2 = a 2 + b 2 cos 90° = 0
Teen sides se angle C cos C = 2 ab a 2 + b 2 − c 2
Kaun se do triangle cases mein shuru karne ke liye Law of Cosines chahiye? SAS (do sides + included angle) aur SSS (teen sides).
Acute angles ke liye correction term negative kyun hoti hai? cos C > 0 − 2 ab cos C < 0 c
Obtuse angle ke liye c cos C < 0 − 2 ab cos C > 0
Coordinate proof mein B B = ( a cos C , a sin C )
a = 8 , b = 5 , C = 60° c c 2 = 64 + 25 − 40 = 49 c = 7
Pythagoras Theorem — C = 90° Law of Sines — iska complement; AAS/ASA aur ambiguous SSA ke liye use hota hai.Dot Product — u ⋅ v = ∣ u ∣∣ v ∣ cos θ Solving Triangles — decision tree: kaunse data ke liye kaun sa law.Unit Circle and Cosine Values — 90° cos
correction term se extend hota hai
distance formula use karta hai
do sides plus included angle
SAS ya SSS shuru nahi kar sakti
