Area of triangle = ½ab·sin C — Hinglish notes

3.1.21 · Maths › Advanced Trigonometry

YEH HAI KYA?

Key condition: angle un dono sides ke beech sandwiched hona chahiye jo aap use kar rahe ho. $C$ side $c$ ke opposite hai, aur yeh sides $a$ aur $b$ ke beech enclosed hai. Isliye yeh $ab$ ke saath aata hai, $a c$ ya $b c$ ke saath nahi.

KYU KAAM KARTA HAI — scratch se derivation

Hum sirf usi area fact se shuru karte hain jis par hum truly trust karte hain: $Area = \frac{1}{2} \times base \times height .$

Step 1 — Base chunno. Side $a$ (vertex $B$ se vertex $C$ tak ki side) ko base lo. Yeh step kyun? Koi kuch base chahiye; $a$ convenient hai kyunki yeh vertex $C$ ko touch karta hai, jahan hamara known angle rehta hai.

Step 2 — Perpendicular height $h$ giraao vertex $A$ se base $a$ wali line par. Yeh step kyun? Height ko chosen base ke perpendicular measure karna hota hai — yahi "height" ki definition hai.

Step 3 — $h$ ko angle $C$ se express karo. Height se bane right-angled triangle ko dekho. Side $b$ ( $C$ se $A$ tak) is chhote right triangle ki hypotenuse hai, aur $h$ angle $C$ ke opposite wali side hai. Sine ki definition se ( $sin = \frac{opp}{hyp}$ ): $sin C = \frac{h}{b} ⟹ h = b sin C .$ Yeh step kyun? Yahi poora point hai — yeh ek unknown height ko known quantities $b$ aur $C$ mein convert karta hai.

Step 4 — Substitute karo. $Area = \frac{1}{2} a h = \frac{1}{2} a (b sin C) = \frac{1}{2} ab sin C .$

KAISE USE KARTE HAIN — worked examples

Common mistakes (steel-manned)

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho ek tent hai jisme do daande hain — ek lamba, ek chhota — jo upar mil ke kuch angle par jhuke hain. Yeh jaanne ke liye ki tent kitna floor cover karta hai, tum base × height chahoge aur use aadha karo (kyunki yeh triangle hai). Lekin tumhare paas height maapne ka ruler nahi! Yahan magic hai: agar tumhe pata ho ki ek daanda kitna lamba hai aur woh kitna jhuka hua hai, toh "woh kitna uuncha pahuncha" bas daande ki length hai times ek special jhukne-wala number jise sine kehte hain. Toh height maapne ki jagah tum use calculate karte ho: height = daanda × sine(angle). Phir wahi purana half of base × height wala kaam. Bas yahi sara raaz hai.

Flashcards

Do sides aur unke included angle se area formula kya hai?

Area = \frac{1}{2} ab sin C

\frac{1}{2} ab sin C

mein

C

kaun sa angle hai?

Woh angle jo sides

a

aur

b

ke beech enclosed (included) hai.

Height

b sin C

kyun hoti hai?

Altitude se bane right triangle mein,

h

angle

C

ke opposite hai aur hypotenuse

b

hai, toh

sin C = h / b \Rightarrow h = b sin C

Area formula mein sine kyun hai cosine kyun nahi?

Height ek perpendicular distance hai = tedhi side ka "across" component = side

\times sin

(angle).

Kya formula obtuse included angle ke liye bhi kaam karta hai?

Haan —

0 < C < 18 0^{\circ}

ke liye

sin C > 0

rehta hai, toh formula valid rehta hai.

Area,

a

b

diye hone par

C

solve karne se do answers kyun aa sakte hain?

Kyunki

sin C = sin (18 0^{\circ} - C)

, ek acute aur ek obtuse solution milta hai (ambiguous case).

C \to 0^{\circ}

hone par area ka kya hoga?

sin C \to 0

, toh area

\to 0

— triangle ek line mein collapse ho jaata hai.

C = 9 0^{\circ}

ke liye formula kya ban jaata hai?

\frac{1}{2} ab

(kyunki

sin 9 0^{\circ} = 1

), legs

a, b

wale standard right-triangle area jaisa.

Connections

Sine Rule — wahi "do sides + angle" family; missing sides/angles dhundhne ke liye use hota hai.
Cosine Rule — teesri side nikalta hai; ek triangle ko puri tarah solve karne ke liye is ke saath pair hota hai.
Heron's Formula — do-sides-ek-angle ki jagah teen sides se area.
Right-Angled Trigonometry (SOHCAHTOA) — $h = b sin C$ ka source.
Cross Product — $∣ u \times v ∣ = ∣ u ∣∣ v ∣ sin θ$ is formula ka vector twin hai.
Unit Circle and Sine — kyun obtuse angle ka $sin$ positive rehta hai.