Trigonometric substitution — x = a sin θ, a tan θ, a sec θ cases: Foundations

4.2.9 · D1 · Maths › Calculus II — Integration › Trigonometric substitution — x = a sin θ, a tan θ, a sec θ c

Parent note ko padhne se pehle bhi, symbols aur pictures ka ek poora stack solid hona chahiye. Yeh page un mein se har ek ko kuch nahi se build karta hai, us order mein jis mein woh ek doosre pe depend karte hain. Agar neeche koi ek bhi link shaky lagey, wahi piece hai jis par slow down karna hai.

0. Woh map jise tum build karne ki taraf ja rahe ho

Taaki baad ke sections ke paas point karne ke liye kuch concrete ho, yahan parent topic ka poora apparatus ek table mein hai. Abhi tumse yeh samajhne ki expectation nahi hai — iske har ek symbol ki definition neeche di gayi hai. Ise paas rakho; sections 3, 7 aur 9 sab specific columns ko refer karte hain.

Definition Ek nazar mein teeno substitutions

Radical	Substitution	$d x$	Root ban jaati hai	Domain of $x$	Range of $θ$
$a^{2} - x^{2}$	$x = a sin θ$	$a cos θ d θ$	$a cos θ$	$∣ x ∣ \leq a$	$θ \in [- \frac{π}{2}, \frac{π}{2}]$
$a^{2} + x^{2}$	$x = a tan θ$	$a sec^{2} θ d θ$	$a sec θ$	all real $x$	$θ \in (- \frac{π}{2}, \frac{π}{2})$
$x^{2} - a^{2}$	$x = a sec θ$	$a sec θ tan θ d θ$	$a ∣ tan θ ∣$	$∣ x ∣ \geq a$	$θ \in [0, π] ∖ {\frac{π}{2}}$

Yahan $a > 0$ hai. Har domain of $x$ column yeh batata hai ki woh triangle model kab apply bhi hota hai; har range of $θ$ column woh interval hai jo root ko $\geq 0$ rakhta hai. Dono derived hain, decreed nahi — sections 3 aur 7 dekho.

1. Right triangle — sab kuch ke neeche wali picture

Is topic mein sab kuch EK picture pe jeeta hai: ek right triangle.

Figure — Trigonometric substitution — x = a sin θ, a tan θ, a sec θ cases

Topic ko isko kyun chahiye: har trig substitution ( $x = a sin θ$ , etc.) asal mein yahi hai — "let $x$ be ek particular side of this triangle." Picture ke bina, substitution ek aisa spell hai jise tum memorise karte ho; iske saath, har step ki ek location hoti hai jis par tum point kar sakte ho.

2. Pythagoras — squares aate hi kyun hain

Yeh kaisa dikhta hai: choti sides par bane do chhote squares ka area milke exactly hypotenuse par bane bade square ko fill karta hai (neeche figure).

Topic ko isko kyun chahiye: teeno ugly radicals $a^{2} - x^{2}$ , $a^{2} + x^{2}$ , $x^{2} - a^{2}$ Pythagoras ko rearrange karke missing side solve karne ke liye hain. For example $a^{2} - x^{2}$ ka matlab hai "hypotenuse $a$ , ek leg $x$ , doosra leg nikalo." Isliye in problems mein hamesha ek triangle chhupa hota hai — Reference right triangle method dekho.

3. — square-root symbol aur uska sign trap

Woh ek rule jo sabko pakadta hai: $u^{2} = ∣ u ∣ (u ki size, sign strip off karke), NOT simply u .$

Topic ko isko kyun chahiye: har derivation ke Step 2 mein hum likhte hain $a^{2} cos^{2} θ = a cos θ$ . Yeh tabhi legal hai jab $cos θ \geq 0$ ho, taaki $∣ cos θ ∣ = cos θ$ . Isko guarantee karna hi poori wajah hai ki har substitution ke saath $θ$ ka ek range aata hai (section-0 table ka aakhri column). Hum woh ranges Section 7 mein carefully build karte hain — abhi ke liye bas note karo: absolute-value bars sirf isliye hatt jaate hain kyunki $θ$ ek aisi interval tak confined hai jahan trig quantity ek hi sign rakhti hai.

4. $a$ aur $x$ — constant vs variable

$a$ kaise nikaalein: woh number hai jo wahan baitha hai jahan $a^{2}$ appear karta hai. $9 - x^{2}$ mein, $a^{2} = 9$ toh $a = 3$ . $x^{2} + 7$ mein, $a^{2} = 7$ toh $a = 7$ .

Topic ko isko kyun chahiye: substitution hamesha $x = a (trig)$ hoti hai. $a$ galat hone se baad ke har step galat scale ho jaate hain.

5. sin, cos, tan — triangle PAR teeno ratios

Ab hum angle $θ$ ko numbers se attach karte hain. Yeh functions hi kyun, kuch aur kyun nahi? Kyunki ek fixed angle $θ$ ke liye, sides ke ratios kabhi nahi badlte chahe triangle kitna bhi bada ho — toh ek ratio angle ka perfect fingerprint hai.

Topic ko isko kyun chahiye: $x = a sin θ$ ka literally matlab hai "let $x$ opposite side ho jab hypotenuse $a$ ho." $x = a tan θ$ ka matlab hai "let $x$ opposite ho jab adjacent $a$ ho." $x = a sec θ$ ka matlab hai "let $x$ hypotenuse ho jab adjacent $a$ ho." Har substitution ek choice hai ki kaunsi sides kaun sa role play karein.

6. Pythagorean identities — woh engines jo root ko khatam karte hain

Neeche ki teeno identities nayi facts nahi hain. Har ek Pythagoras hai, $sin^{2} θ + cos^{2} θ = 1$ , jo ek alag quantity se divide kiya gaya hai. Yahan poori derivation hai, ek ek line. Zyada ke liye Pythagorean identities dekho.

Topic ko isko kyun chahiye — poora trick ek line mein: root ke neeche do squares ka sum ya difference mushkil hai, lekin root ke neeche ek akela square trivial hai ( $(stuff)^{2} = ∣ stuff ∣$ ). Yeh identities do terms ko ek squared term mein collapse kar deti hain. Wahi payoff hai, aur isliye hum koi random change of variable ki jagah trig substitutions choose karte hain.

7. arcsin, arctan, arcsec — ratio ko undo karna, aur ranges kahan se aati hain

Range kyun zaroori hai: $sin θ$ hamesha ke liye repeat karta hai, toh "kaun sa angle" ke infinitely many answers hain. Hum ek ko pin down karte hain $θ$ restrict karke. Hum woh restriction iss tarah choose karte hain ki (a) $x$ ka poora domain cover ho jaaye aur (b) root $\geq 0$ rahe. Yahan har case apni reasoning ke saath hai:

$sin$ -case, $x = a sin θ$ . Kyunki $- 1 \leq sin θ \leq 1$ , hume milta hai $- a \leq x \leq a$ , yani domain $∣ x ∣ \leq a$ . $θ \in [- \frac{π}{2}, \frac{π}{2}]$ choose karne se $sin θ$ exactly ek baar $[- 1, 1]$ ke across sweep hota hai, aur us interval par $cos θ \geq 0$ , toh $a^{2} - x^{2} = a cos θ$ (no bars).
$tan$ -case, $x = a tan θ$ . Jab $θ$ , $(- \frac{π}{2}, \frac{π}{2})$ par run karta hai, $tan θ$ har real value ek baar leta hai, toh domain all real $x$ hai. Us interval par $sec θ > 0$ , toh $a^{2} + x^{2} = a sec θ$ (no bars).
$sec$ -case, $x = a sec θ$ . Kyunki $∣ sec θ ∣ \geq 1$ , hume milta hai $∣ x ∣ \geq a$ , yani domain $∣ x ∣ \geq a$ . Standard choice hai $θ \in [0, π] ∖ {\frac{π}{2}}$ : $[0, \frac{π}{2})$ par hum $sec θ \geq 1$ (toh $x \geq a$ ) cover karte hain, aur $(\frac{π}{2}, π]$ par hum $sec θ \leq - 1$ (toh $x \leq - a$ ) cover karte hain. Caution: $(\frac{π}{2}, π]$ par $tan θ \leq 0$ hota hai, toh $x^{2} - a^{2} = a ∣ tan θ ∣ = - a tan θ$ wahan. Woh sign flip (section-0 table mein $a ∣ tan θ ∣$ ke roop mein dikhaya gaya) exactly wajah hai ki $sec$ -case ko negative $x$ ke liye extra care chahiye.

Topic ko isko kyun chahiye: har problem ka final answer wapas $x$ mein hona chahiye. Kyunki $x = a sin θ$ deta hai $sin θ = \frac{x}{a}$ , angle khud hai $θ = arcsin \frac{x}{a}$ . Inverse trig hi ek tarika hai $θ$ ko $x$ ke terms mein likhne ka.

8. $\int f d x$ aur $d x$ — integral aur uski tail

Topic ko isko kyun chahiye: jab hum $x = a sin θ$ set karte hain, width $d x$ ko $θ$ mein rewrite karna padta hai. Hum yeh differentiate karke karte hain (Section 9). $d x$ conversion bhool jaana trig-sub ki sabse common error hai.

9. Differential $d x = g^{'} (θ) d θ$ — width kaise transform hoti hai

Tumhe sirf teeno derivative facts chahiye, har case ke liye ek (har ek standard rule hai, neeche verify kiya gaya): $x = a sin θ \Rightarrow d x = a cos θ d θ, x = a tan θ \Rightarrow d x = a sec^{2} θ d θ, x = a sec θ \Rightarrow d x = a sec θ tan θ d θ .$

Topic ko isko kyun chahiye: yahi Integration by substitution (u-sub) ki machinery hai — trig substitution ek u-sub hai jahan new variable ek angle hota hai. Transformed $d x$ usually simplified root ke against beautifully cancel ho jaata hai, jo woh moment hai jab integral doable ban jaata hai.

θ

ke relative naam batao.

Hypotenuse (right angle ke opposite), opposite (

θ

ke across), adjacent (

θ

ko touch karne wali).

Right triangle ke liye Pythagoras state karo.

(opp)^{2} + (adj)^{2} = (hyp)^{2}

u^{2}

kya equal hai — aur topic ko isse kyun care hai?

∣ u ∣

;

θ

-range inside ko non-negative banata hai toh bars hatt jaate hain.

x^{2} + 7

mein

a

kya hai?

a = 7

(kyunki

a^{2} = 7

sin θ

cos θ

tan θ

ko side ratios ke roop mein likho.

sin θ = opp / hyp

cos θ = adj / hyp

tan θ = opp / adj

sec θ

cos θ

ke terms mein kya hai?

sec θ = 1/ cos θ

Pythagoras se

tan^{2} θ + 1 = sec^{2} θ

derive karo.

sin^{2} θ + cos^{2} θ = 1

cos^{2} θ

se divide karo.

Har substitution ke liye

x

ka domain batao.

∣ x ∣ \leq a

for

a sin θ

; all real

x

for

a tan θ

;

∣ x ∣ \geq a

for

a sec θ

sec

-case ke liye

θ

ka standard range batao.

θ \in [0, π] ∖ {\frac{π}{2}}

arcsin (r)

kaun sa sawaal answer karta hai?

"Kaun sa angle

θ

hai jiska

sin θ = r

ho?" (apne range ke andar).

Derivative

g^{'} (θ)

ki definition state karo.

g^{'} (θ) = lim_{h \to 0} \frac{g ( θ + h ) - g ( θ )}{h}

, instantaneous slope.

Agar

x = a tan θ

, toh

d x

kya hai?

d x = a sec^{2} θ d θ

Integration ke liye

cos^{2} θ

rewrite karo.

cos^{2} θ = \frac{1 + cos 2 θ}{2}

\int f d x

mein

d x

optional kyun nahi hai?

Yeh slice width hai; variable change karne se yeh change hota hai, toh isko convert karna padta hai.

Connections

4.2.09 Trigonometric substitution — x = a sin θ, a tan θ, a sec θ cases (Hinglish) — parent topic, Hinglish version.
Pythagorean identities — woh engine jo root ko khatam karta hai.
Integration by substitution (u-sub) — trig sub ek u-sub hai disguise mein.
Power-reduction & double-angle formulas — $\int cos^{2} θ$ ke liye finisher.
Reference right triangle method — Section 1 ki picture back-substitution ko kaise power deti hai.
Hyperbolic substitution — cousin method $cosh, sinh$ use karta hai.

Foundations — Trigonometric substitution — x = a sin θ, a tan θ, a sec θ cases