Law of sines — proof and applications (ambiguous case)

3.1.19 · Maths › Advanced Trigonometry

Law of Sines KYA hai?

Cloze karne wala key term: common ratio == $2 R$ == ke barabar hota hai (circumdiameter).

Yeh sach KYU hai? — Scratch se Derivation

Step 1 — Ek altitude drop karo (core idea)

Triangle $A B C$ lo. Vertex $C$ se side $A B$ par height $h$ ka perpendicular daalo.

Yeh step kyu? Ek altitude triangle ko do right triangles mein split kar deta hai, aur ek right triangle mein $sin (angle) = \frac{opposite}{hypotenuse}$ — yahi ek trig fact humhe chahiye.

Left wale right triangle mein, angle $A$ ki opposite side $h$ hai aur hypotenuse $b$ hai: $sin A = \frac{h}{b} ⟹ h = b sin A$

Right wale right triangle mein, angle $B$ ki opposite side $h$ hai aur hypotenuse $a$ hai: $sin B = \frac{h}{a} ⟹ h = a sin B$

Yeh step kyu? Dono expressions usi same height $h$ ke barabar hain, toh hum unhe equate kar sakte hain.

$b sin A = a sin B ⟹ \frac{a}{s i n A} = \frac{b}{s i n B}$

$A$ se $B C$ par altitude daalne par $\frac{b}{sin B} = \frac{c}{sin C}$ bhi mil jaata hai. Ratio equality ke liye ho gaya.

Step 2 — Ratio $2 R$ KYU hai (circumcircle part)

Radius $R$ ka circumcircle draw karo. Vertex $B$ se diameter $B D$ draw karo (toh $D$ circle par hai, $B D = 2 R$ ).

Angle $∠ B C D = 90°$ (semicircle mein angle right angle hota hai).
Angles $∠ A$ aur $∠ D$ same chord $B C$ ko subtend karte hain, toh inscribed-angle theorem se $∠ D = ∠ A$ .

Right triangle $B C D$ mein: $sin D = \frac{B C}{B D} = \frac{a}{2 R}$

Kyunki $sin D = sin A$ : $sin A = \frac{a}{2 R} ⟹ \frac{a}{s i n A} = 2 R$

Yeh step kyu? Yeh ek sirf "equal ratios" wale statement ko ek geometric constant mein upgrade kar deta hai: ratio literally usi circle ko measure karta hai jisme triangle rehta hai.

Figure — Law of sines — proof and applications (ambiguous case)

Hum ise kaise use karte hain? (Kab use karna hai)

Law of Sines tab use karo jab tumhare paas ho:

AAS / ASA — do angles aur ek side → baaki find karo (unique, safe).
SSA — do sides aur ek angle jo unke beech nahi — ambiguous case ⚠️.

Ambiguous case, decoded (maano $A$ acute hai, $a, b, A$ diya gaya hai)

$sin B = \frac{b sin A}{a}$ calculate karo. Critical length altitude $h = b sin A$ hai.

Condition	Triangles ki sankhya
$a < h = b sin A$	0 (side itni choti ki base tak pahunch hi nahi sakti)
$a = h$	1 (right triangle, bas touch karta hai)
$h < a < b$	2 (arc base ko do baar kaata hai: $B$ aur $180° - B$ )
$a \geq b$	1 (lambi side, sirf ek valid $B$ )

" $180° - B$ " doosra solution kyu? Kyunki $sin B = sin (180° - B)$ , $B$ ki equation ke $(0°, 180°)$ mein do answers ho sakte hain. Doosra sirf tab rakho jab $A + (180° - B) < 180°$ ho, yani angles abhi bhi triangle mein fit hon.

Worked examples

Recall 12-saal ke bachche ko explain karo (Feynman)

Socho ek triangle ek circle ke andar perfectly draw hua hai. Law of Sines kehta hai: koi bhi side lo, usse uske saamne wale corner ke "sine" se divide karo — tumhe hamesha same number milega, aur woh number exactly itna wide hoga jitna poora circle hai (uska diameter). Tricky "ambiguous" part yeh hai: maano main tumhe ek corner ka angle aur do stick lengths batata hun, lekin exactly kaise connect hote hain nahi batata. Yeh ek door hinge jhulane jaisa hai — kabhi kabhi door ka tip wall se do jagah takraata hai, kabhi ek jagah, kabhi reach hi nahi kar paata. Toh do alag triangles ho sakte hain, ek, ya koi bhi nahi!

Common mistakes

Active-recall flashcards

#flashcards/maths

Common ratio

\frac{a}{s i n A}

geometrically kiske barabar hai?

Circumdiameter

2 R

(circumcircle radius ka double).

Law of Sines state karo.

\frac{a}{s i n A} = \frac{b}{s i n B} = \frac{c}{s i n C} = 2 R

Proof mein,

h = b sin A

tab bhi kyu hold karta hai jab

A

obtuse ho?

Kyunki

sin (180° - A) = sin A

, toh altitude relation unchanged rehta hai.

Law of Sines kaun se triangle-inference cases mein use hota hai?

AAS, ASA (unique) aur SSA (ambiguous).

Proof mein

2 R

part kaun sa theorem deta hai?

Inscribed-angle theorem aur "semicircle mein angle

90°

hota hai".

a, b, A

diya gaya ho (A acute), critical altitude length kya hai?

h = b sin A

SSA exactly do triangles kab deta hai?

Jab

h < a < b

ho (yani

b sin A < a < b

SSA kab koi triangle nahi deta?

Jab

a < b sin A

ho, equivalently

sin B > 1

SSA exactly ek triangle kab deta hai (medium/long)?

Jab

a = b sin A

ho (right angle, tangent) ya

a \geq b

ho.

Kabhi kabhi doosra solution

180° - B

kyu hota hai?

Kyunki

sin B = sin (180° - B)

; ise tabhi rakho jab

A + (180° - B) < 180°

ho.

SAS ya SSS ke liye kaun sa law use karna chahiye?

Law of Cosines.

Connections

Law of Cosines — SAS/SSS ke liye go-to; SSA mein Sines ko complement karta hai.
Inscribed Angle Theorem — $2 R$ result ko power deta hai.
Circumcircle and Circumradius R — ratio ka geometric meaning.
Sine of Supplementary Angles — $sin (180° - θ) = sin θ$ , ambiguity ki jad.
Area of a Triangle — $Area = \frac{ab c}{4 R}$ yahan $R$ se directly link karta hai.
Solving Oblique Triangles — decision flowchart AAS/ASA/SSA/SAS/SSS.