🏀 *त्रिकोणमिति : महत्वपूर्ण सूत्र (IMP Formula)* 🏀

💠 *योग सूत्र* 😀

🔹 Sin(A + B) = SinACosB +
      CosASinB
🔺Sin(A - B) = SinACosB -
      CosASinB
🔹Cos(A + B) = CosACosB -
      SinASinB
🔺Cos(A - B) = CosA.CosB +
      SinA.SinB

💠 *अन्तर सूत्र* 😀

🔹tan(A + B) = tanA + tanB/1 -
      tanA.tanB
🔺tan(A - B) = tanA - tanB/1 +
      tanA.tanB

💠 *C - D सूत्र* 😀

🔹SinC + SinD = 2Sin(C + D/2)
      Cos(C - D/2)
🔺SinC - SinD = 2Cos(C + D/2)
      Sin(C - D/2)
🔹CosC + CosD = 2Cos(C + D/
      2) Cos(C - D/2)
🔺CosC - CosD = 2Sin(C + D/2)
      Sin(D - C/2)
🔹CosC - CosD = - 2Sin(C + D/
      2) Sin(C - D/2)

💠 *रूपांतरण सूत्र 😀*

🔹2SinA.CosB = Sin(A + B) +
      Sin(A-B)
🔺2CosA.SinB = Sin(A + B) -
      Sin(A - B)
🔹2CosA.CosB = Cos(A + B) +
      Cos(A-B)
🔺2SinA.SinB = Cos(A - B) -
      Cos(A + B)

💠 *द्विक कोण सूत्र* 😀

🔹Sin2A = 2SinA.CosA
🔺Cos2A = Cos²A - Sin²A =
      2Cos² - 1 = 1 - 2Sin²A
🔹tan2A = 2tanA/1 - tan²A
🔺Sin2A = 2tanA/1 + tan²A
🔹Cos2A = 1 - tan²A/1 + tan²A

💠 *विशिष्ट सूत्र* 😀

🔹Sin(A + B) = Sin A . Cos B +
      Cos A . Sin B
🔺Sin(A - B) = Sin A . Cos B −
      Cos A . Sin B
🔹Cos (A + B) = Cos A . Cos B −
      Sin A . Sin B
🔺Cos ( A - B ) = Cos A . Cos B
      + Sin A . Sin B

💠 *त्रिक कोण सूत्र* 😀

🔹Sin3A = 3SinA - 4Sin³A
🔺Cos3A = 4Cos³A - 3CosA
🔹tan3A = 3tanA - tan³A/1 - 3tan²A

💠 *महत्वपूर्ण सर्वसमिकाएं* 🔹

🔹Sin²θ + Cos²θ = 1
🔺Sin²θ = 1 - Cos²θ
🔹Cos²θ = 1 - Sin²θ
🔺1 + tan²θ = Sec²θ
🔹Sec²θ - tan²θ = 1
🔺tan²θ = Sec²θ - 1
🔹1 + Cot²θ = Cosec²θ
🔺Cosec²θ - Cot²θ = 1
🔹Cot²θ = Cosec²θ - 1

ALGEBRA

Formulas importantes

*🔢 वर्ग और वर्गमूल तालिका (1-30)*

*🔷 Squares (वर्ग):*
1² = 1
2² = 4
3² = 9
4² = 16
5² = 25
6² = 36
7² = 49
8² = 64
9² = 81
10² = 100
11² = 121
12² = 144
13² = 169
14² = 196
15² = 225
16² = 256
17² = 289
18² = 324
19² = 361
20² = 400
21² = 441
22² = 484
23² = 529
24² = 576
25² = 625
26² = 676
27² = 729
28² = 784
29² = 841
30² = 900
*🔶 Square Roots (वर्गमूल):*
√1 = 1
√4 = 2
√9 = 3
√16 = 4
√25 = 5
√36 = 6
√49 = 7
√64 = 8
√81 = 9
√100 = 10
√121 = 11
√144 = 12
√169 = 13
√196 = 14
√225 = 15
√256 = 16
√289 = 17
√324 = 18
√361 = 19
√400 = 20
√441 = 21
√484 = 22
√529 = 23
√576 = 24
√625 = 25
√676 = 26
√729 = 27
√784 = 28
√841 = 29
√900 = 30

*📍 Trick: 11² = 121, 13² = 169, 17² = 289 — ये Exams में बार-बार पूछे जाते हैं।*

*Please hit like💓& share*

✅10TH और 12TH आल❄️ सब्जेक्ट पीडीऍफ़,🦋नोट्स MCQ 👇👇👇
━━━━━━✧❂✧━━━━━━
https://t.me/addlist/zlRzQkFgOY1mZGVl

*🔢 वर्ग और वर्गमूल तालिका (1-30)*

*🔷 Squares (वर्ग):*
1² = 1
2² = 4
3² = 9
4² = 16
5² = 25
6² = 36
7² = 49
8² = 64
9² = 81
10² = 100
11² = 121
12² = 144
13² = 169
14² = 196
15² = 225
16² = 256
17² = 289
18² = 324
19² = 361
20² = 400
21² = 441
22² = 484
23² = 529
24² = 576
25² = 625
26² = 676
27² = 729
28² = 784
29² = 841
30² = 900
*🔶 Square Roots (वर्गमूल):*
√1 = 1
√4 = 2
√9 = 3
√16 = 4
√25 = 5
√36 = 6
√49 = 7
√64 = 8
√81 = 9
√100 = 10
√121 = 11
√144 = 12
√169 = 13
√196 = 14
√225 = 15
√256 = 16
√289 = 17
√324 = 18
√361 = 19
√400 = 20
√441 = 21
√484 = 22
√529 = 23
√576 = 24
√625 = 25
√676 = 26
√729 = 27
√784 = 28
√841 = 29
√900 = 30

*📍 Trick: 11² = 121, 13² = 169, 17² = 289 — ये Exams में बार-बार पूछे जाते हैं।*

*Please hit like💓& share*

⬇️10th/12th. का नोट्स डाउनलोड कर लो सब ⬇️
https://nephobox.com/s/16I3TlBMBtRD3LGnd1BkDmwp

12th classs vvi formula 👇


Relations and Functions (सम्बंध और फलन)

Domain: Input values (x)

Range: Output values (y)

Inverse Function: f⁻¹(y) = x

Composition: (f ∘ g)(x) = f(g(x))

🧮 2. Inverse Trigonometric Functions

Formula Value

sin⁻¹(–x) = –sin⁻¹(x) cos⁻¹(–x) = π – cos⁻¹(x)
tan⁻¹(–x) = –tan⁻¹(x) cot⁻¹(–x) = π – cot⁻¹(x)
sin⁻¹x + cos⁻¹x = π/2 tan⁻¹x + cot⁻¹x = π/2

🧮 3. Matrices (आव्यूह)

Addition: (A + B)ᵀ = Aᵀ + Bᵀ

Multiplication: (AB)ᵀ = BᵀAᵀ

Identity Matrix: A·I = A

Inverse: A⁻¹ = (1/|A|)·Adj(A)

🧮 4. Determinants (सारक)

For 2×2 Matrix:
|a b|
|c d| = ad – bc

For 3×3 Matrix:
|a₁ b₁ c₁|
|a₂ b₂ c₂|
|a₃ b₃ c₃| = a₁(b₂c₃–b₃c₂) – b₁(a₂c₃–a₃c₂) + c₁(a₂b₃–a₃b₂)

🧮 5. Continuity & Differentiability

d/dx (xⁿ) = n·xⁿ⁻¹

d/dx (sinx) = cosx

d/dx (cosx) = –sinx

d/dx (tanx) = sec²x

d/dx (eˣ) = eˣ

d/dx (lnx) = 1/x

Chain Rule → d/dx [f(g(x))] = f′(g(x))·g′(x)

🧮 6. Applications of Derivatives

Rate of Change: dy/dx

Slope of Tangent: m = dy/dx

Maxima/Minima: f′(x)=0, check f′′(x)

Increasing Function: f′(x) > 0

Decreasing Function: f′(x) < 0

🧮 7. Integrals (समाकलन)

Basic Formulas:

1. ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C

2. ∫ 1/x dx = ln|x| + C

3. ∫ eˣ dx = eˣ + C

4. ∫ sinx dx = –cosx + C

5. ∫ cosx dx = sinx + C

6. ∫ sec²x dx = tanx + C

7. ∫ csc²x dx = –cotx + C

8. ∫ 1/(1 + x²) dx = tan⁻¹x + C