Solving trig equations — general solutions, solutions in given range

3.1.18 · Maths › Advanced Trigonometry

1. Infinitely many solutions KYUN? (first principles)

YEH solutions kaise create karta hai: agar $θ_{0}$ equation solve karta hai, toh periods ke kisi bhi whole number ko add karne se ek aur solution milta hai, kyunki function usi value par wapas aa jaata hai.

Lekin sine aur cosine dono ek value ko do jagah hit karte hain har cycle mein (peaks ke alawa). Toh humein unke liye do families of angles chahiye, tan ke liye ek family.

2. General solutions derive karna

2.1 Cosine — $cos θ = c$

$α = cos^{- 1} c$ ko principal angle maano. Ek turn mein do solutions hain $θ = α$ aur $θ = - α$ (yaani $2 π - α$ ). Period $2 π$ ke multiples add karo:

$\pm$ kyun? even symmetry ki wajah se. $2 nπ$ kyun? kyunki yahi period hai.

2.2 Sine — $sin θ = s$

$α = sin^{- 1} s$ maano. Toh:

Check karo ki kaam karta hai: $n = 0 \Rightarrow α$ . $n = 1 \Rightarrow π - α$ . $n = 2 \Rightarrow 2 π + α$ . Exactly sine solutions. ✓

2.3 Tangent — $tan θ = t$

$α = tan^{- 1} t$ maano.

3. Ek given range mein solutions kaise dhundhe (recipe)

4. Worked examples

5. Forecast-then-Verify

Recall Answer padhne se pehle predict karo:

$sin θ = 0.4$ ke $0 \leq θ < 2 π$ mein kitne solutions hain? $0 \leq θ < 4 π$ mein? $2 π$ turn mein $2$ solutions → pehli range mein $2$ , doosri mein $4$ . General formula confirm karta hai: $n$ mein 2 ka har increment ek poora period-pair complete karta hai.

6. Common mistakes (Steel-manned)

7. Feynman

Recall Ek 12-saal ke bacche ko explain karo

Socho ek Ferris wheel hai jo forever ghhumti rehti hai. Agar main poochhun "tumhari seat exactly 3 metre upar kab hogi?", toh sirf ek time nahi hoga — yeh har loop mein hota hai, har loop mein do baar (ek baar upar jaate waqt, ek baar neeche aate waqt). General solution kuch aisa kehne jaisa hai: "in do clock positions par, aur phir har full turn ke baad dobara." Chota number $n$ bas yeh batata hai ki tum kaunse loop par ho. Agar tumhe sirf lunch se pehle ke answers chahiye, toh sirf woh loops rakho jo lunch se pehle khatam hote hain.

8. Connections

Unit Circle and Radian Measure — jahaan se $α$ aur $\pm$ aate hain.
Graphs of Trigonometric Functions — periodicity ka visual aur kitni roots hain.
Inverse Trigonometric Functions — principal value $α$ .
Trigonometric Identities — complex equations ko $sin / cos / tan = c$ mein reduce karo.
Compound and Double Angle Formulae — multiple-angle equations ke liye zaroori.

General solution of

cos θ = cos α

θ = 2 nπ \pm α, n \in Z

General solution of

sin θ = sin α

θ = nπ + (- 1)^{n} α, n \in Z

General solution of

tan θ = tan α

θ = nπ + α, n \in Z

Cosine ke general solution mein

\pm

kyun hota hai?

Kyunki

cos

even hai,

cos (- α) = cos α

, har turn mein do symmetric roots deta hai.

Tangent ko sirf ek family kyun chahiye?

Iska period

π

hai aur yeh har period mein har value exactly once hit karta hai.

cos 2 θ = c

0 \leq θ < 2 π

par solve karte waqt pehle kya change karna chahiye?

Range ko

0 \leq 2 θ < 4 π

tak widen karo solve karne se pehle, phir results ko 2 se divide karo.

sin θ = 0.4

0 \leq θ < 2 π

mein kitne solutions hain?

Do.

Integer

n

physically kya represent karta hai?

Kaun sa poora period (loop) jisme solution hai.

Kisi bhi trig equation mein pehla step kya hai?

sin / cos / tan

of the angle ko ek number ke barabar isolate karne ke liye rearrange karo.