Polar form — r(cos θ + i sin θ) = r·cis θ

3.5.5 · Maths › Complex Numbers

WHAT hai polar form?

HOW derive karte hain? (scratch se)

Ek point $z = a + bi$ ko Argand plane par plot karo. Origin $O$ se us point tak arrow draw karo. Uski length $r$ rakho aur positive real axis se jo angle banta hai usse $θ$ rakho.

Step 1 — right triangle banao. Point se real axis par ek perpendicular drop karo. Horizontal leg $= a$ , vertical leg $= b$ , hypotenuse $= r$ . Ye step kyun? Trigonometry sirf right triangles par kaam karti hai, isliye hum ek manually bana lete hain.

Step 2 — trig ratios padho. $cos θ = \frac{adjacent}{hyp} = \frac{a}{r}, sin θ = \frac{opposite}{hyp} = \frac{b}{r}$ Ye step kyun? Ye literally angle $θ$ ke liye cos aur sin ki definitions hain.

Step 3 — $a$ aur $b$ ke liye solve karo. $a = r cos θ, b = r sin θ$ Ye step kyun? Hum $a, b$ ko $r, θ$ se replace karna chahte hain, isliye unhe subject banao.

Step 4 — $z = a + bi$ mein substitute karo. $z = r cos θ + i (r sin θ) = r (cos θ + i sin θ)$ Ho gaya — yahi polar form hai, purely Pythagoras + trig se derive kiya. ∎

Quadrant trap (WHY $tan^{- 1}$ akela kaafi nahi)

Calculator ka $tan^{- 1} (b / a)$ sirf $(- 90°, 90°)$ mein angles return karta hai. Lekin arrows $360°$ mein kahin bhi point kar sakte hain. Isliye tumhe dekhna hoga ki $(a, b)$ kis quadrant mein hai aur accordingly adjust karna hoga.

| Quadrant | Signs $(a, b)$ | Reference angle $α = tan^{- 1} \frac{b}{a}$ se argument | |---|---|---| | I | $(+, +)$ | $θ = α$ | | II | $(-, +)$ | $θ = π - α$ | | III | $(-, -)$ | $θ = - (π - α)$ ya $π + α$ | | IV | $(+, -)$ | $θ = - α$ |

Principal argument $(- π, π]$ mein choose kiya jaata hai.

Worked examples

Forecast-then-verify

Recall Check karne se pehle predict karo:

$ar g (- 1)$ aur $ar g (- i)$ kya hai? Forecast: $- 1$ negative real axis par baitha hai; $- i$ seedha neeche point karta hai. Verify: $ar g (- 1) = π$ (arrow left point karta hai). $ar g (- i) = - \frac{π}{2}$ (principal value; neeche point karta hai). Dono ka modulus $1$ hai. Toh $- 1 = cis π$ aur $- i = cis (- \frac{π}{2})$ .

Common mistakes (steel-manned)

Flashcards

cis θ

ka matlab kya hai?

cos θ + i sin θ

z = a + bi

ke modulus ka formula?

r = a^{2} + b^{2}

Argument ke liye sirf

tan^{- 1} (b / a)

kyun use nahi kar sakte?

Ye quadrant ignore karta hai;

tan

ka period

π

hai, isliye

180°

apart points same value dete hain. Tumhe

(a, b)

ke quadrant se adjust karna hoga.

1 + i

ko polar form mein convert karo.

2 cis \frac{π}{4}

2 cis \frac{2 π}{3}

ko Cartesian mein convert karo.

- 1 + i 3

Kya modulus

r

negative ho sakta hai?

Nahi,

r \geq 0

hamesha; direction

θ

mein carry hota hai.

Polar form mein do complex numbers multiply karne ka rule?

Moduli multiply karo, arguments add karo.

- i

ka argument (principal value)?

- \frac{π}{2}

Quadrant II mein, reference angle

α

θ

kaise milta hai?

θ = π - α

z = a + bi

kaunsa geometric object represent karta hai?

Argand (complex) plane mein ek point/arrow.

Recall Feynman: ek 12-saal ke bacche ko samjhao

Socho tum ek bade maidan ke beech mein khade ho. Kahin khazana gada hua hai. Apne dost ko batane ka ek tarika: "3 kadam East, phir 4 kadam North chalo" — ye $3 + 4 i$ jaisa hai. Doosra tarika: "5 kadam chalo, us direction mein" — tum sahi angle par ghoomte ho aur chalte ho. Ye "5 kadam + ek direction" polar form hai. Kadam ki sankhya modulus $r$ hai, aur jo direction tum face kar rahe ho wo angle $θ$ hai. Sama khazana, do alag tarike se describe kiya. Polar form tab kaam aata hai jab arrows ko spin aur stretch karna ho — sirf angles add karo aur lengths multiply karo.

Connections

Argand Diagram — wo plane jahan $r$ aur $θ$ rehte hain.
Modulus and Argument — polar form ke do ingredients.
Multiplication and Division of Complex Numbers — jahan polar form fayda deta hai (angles add/subtract karo).
De Moivre's Theorem — $(cis θ)^{n} = cis n θ$ , seedha isi par built hai.
Euler's Formula — $e^{i θ} = cos θ + i sin θ$ , yahi deeper reason hai ki cis exponent ki tarah behave karta hai.
Roots of Complex Numbers — $n$ -th roots nikalne ke liye polar form use karta hai.