3.1.2
Ek angle ko circle par kata hua arc ka length se measure kiya ja sakta hai — poore ghoom ko arbitrarily 360 parts mein baantne se nahi. Ek radian angle ko seedha circle ki apni geometry se jodata hai: "maine kitne radius-length ke arc sweep kiye?" Yeh natural hai kyunki iska matlab hai ki arc length aur area jaise formulas saaf aate hain, koi extra conversion factors nahi.
Degrees mein kya problem hai?
Degrees ek human convention hai. Babylonians ne 360 choose kiya (achhe divisors, ~saal mein din). Circle ko 360 "pata" nahi hota. Isliye degrees use karne wale formulas mein 180 π
Radians kyun theek hain: angle ko circle ki khud ki geometry se define karo.
Ek radian woh angle hai jo ek circle ke centre par tab banta hai jab arc ki length circle ki radius ke barabar ho.
θ ( in radians ) = radius r arc length s
Yeh ek pure number kyun hai: θ = s / r dimensionless hote hain. Isliye hum aksar "rad" unit drop kar dete hain.
Intuition Sab kuch EK fact par anchor karoPoore circle ki circumference C = 2 π r Poora ghoom 36 0 ∘
Step 1 — Full turn radians mein. Kyun? Poora sweep matlab arc poori circumference hai, s = 2 π r θ full = r s = r 2 π r = 2 π radians
Step 2 — Usi full turn ke dono measures ko equate karo. Kyun? Same physical angle, do labels.
36 0 ∘ = 2 π rad ⟹ 18 0 ∘ = π rad
Step 3 — Conversion padho. Kyun? "1 ∘ 1
Numerically: 1 rad = π 180 ≈ 57.295 8 ∘ 6 0 ∘
Kyunki θ = s / r bane the:
6 0 ∘ 180 π Kyun? Degrees→radians factor.
60 × 180 π = 180 60 π = 3 π rad Check: 3 π ≈ 1.047 1.047 × 57.3 ≈ 6 0 ∘
4 3 π π 180 Kyun? Radians→degrees factor.
4 3 π × π 180 = 4 3 × 180 = 13 5 ∘ π π π
Worked example 3) Arc length: circle radiusr = 5 4 0 ∘ Step A — radians mein convert karo (formula s = r θ radians maangta hai ).
θ = 40 × 180 π = 9 2 π rad Step B — s = r θ Kyun? Definition rearranged.
s = 5 × 9 2 π = 9 10 π ≈ 3.49 cm
Worked example 4) Forecast-then-VerifyForecast: Kya 2 rad 9 0 ∘ Pehle guess karo.
Verify: 2 × π 180 ≈ 114. 6 ∘ 9 0 ∘ zyada — kyunki 1 rad ≈ 57° hota hai, do ≈ 114°. ✓
s = r θ Kyun sahi lagta hai: "θ Kyun galat hai: s = r θ θ = s / r radian define karta hai. Degrees mein s = 180 π r θ d e g Fix: s = r θ A = 2 1 r 2 θ
Common mistake Galat factor se multiply karnaKyun sahi lagta hai: 180 π π 180 Fix / sanity check: radians chhote numbers hote hain, degrees bade . Deg→rad jaate waqt number shrink hona chahiye (180 π < 1 badhna chahiye.
Common mistake Radians ka dimensionless hona bhool jaana"rad" likhna jaise yeh metres ho. Fix: sin θ θ d θ d sin θ = cos θ
Recall Feynman: ek 12-saal ke bacche ko explain karoSocho tum ek gol talab ke chaaron taraf chal rahe ho. "Maine 90 degrees muda" kehne ki jagah tum kaho "maine ek talab-radius jitna kinara chala." Radian exactly yahi hai: tumne kitne radius-sticks kinare chale. Poora ghoomne par tumne 2 π 18 0 ∘ π 18 0 ∘ = π
Mnemonic Factor yaad rakhna"180 = π" master key hai. Convert karne ke liye, jo unit chahiye use upar rakho:
radians chahiye → × 180 π × π 180 "ek radian ~57 hota hai, toh yeh BADA-ish angle hai."
Ek radian ki definition Woh angle jo circle ke centre par tab banta hai jab arc ki length radius ke barabar ho.
Arc aur radius ke terms mein angle radians mein nikalne ka formula θ = s / r
Poore circle mein kitne radians hote hain? 2 π = 36 0 ∘
Conversion ke liye master identity 18 0 ∘ = π
Degrees → radians convert karne ka factor π /180
Radians → degrees convert karne ka factor 180/ π
1 radian ki approximate value degrees mein ≈ 57. 3 ∘
9 0 ∘ π /2
π /6 3 0 ∘
Arc length formula (θ rad mein) s = r θ
Sector area formula (θ rad mein) A = 2 1 r 2 θ
s = r θ Kyunki s = r θ θ = s / r
Kya radians ek dimension wali unit hai? Nahi — length/length, isliye dimensionless (pure number) hai.
rearrange s equals r theta
Dimensionless pure number
Full circle C equals 2 pi r
Full turn equals 2 pi rad
Sector area half r squared theta
