🔸 Ratios & Identities

sin θ = opp/hyp, cos θ = adj/hyp, tan θ = sin/cos

Key identities: sin²θ + cos²θ = 1, 1+tan²θ = sec²θ

📈 Graphs

sinθ (red) & cosθ (blue) – periodic, amplitude 1

⚡ Differentiation

d/dx (xⁿ) = n xⁿ⁻¹    d/dx (sin x) = cos x

d/dx (cos x) = -sin x    d/dx (eˣ) = eˣ

📦 Integration

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C   (n≠-1)

∫ sin x dx = -cos x + C   ∫ cos x dx = sin x + C

Area under v-t graph = displacement

➕ Vector Addition

Triangle law: R = A + B (head-to-tail)

Magnitude: |R| = √(A² + B² + 2AB cos θ)

Direction: tan α = (B sin θ)/(A + B cos θ)

➖ Subtraction

A – B = A + (-B)   (reverse direction of B)

|A – B| = √(A² + B² – 2AB cos θ)

Any vector can be broken into components: A = Aₓ î + Aᵧ ĵ + A₂ k̂

Aₓ = A cos θ, Aᵧ = A sin θ (2D)

Magnitude: |A| = √(Aₓ² + Aᵧ² + A₂²)

🔵 Dot (Scalar) Product

A·B = |A||B| cos θ = AₓBₓ + AᵧBᵧ + A₂B₂

Result: scalar. Used for work (W = F·d).

🟢 Cross (Vector) Product

|A×B| = |A||B| sin θ, direction ⊥ to plane (right-hand rule).

A×B = (AᵧB₂ – A₂Bᵧ)î + ... (determinant)

💡 NEET TIPS & TRICKS

• sin θ ≈ θ (radians) for small angles – used in SHM.
• Maximum resultant: when θ=0°, |A+B|; minimum: θ=180°, |A–B|.
• For vector addition, always use component method to avoid mistakes.
• Unit vector along direction: (position vector)/magnitude.

⚠️ COMMON MISTAKES

• Confusing scalar vs vector: pressure, energy are scalar.
• Forgetting direction while adding vectors (use head-to-tail).
• Misusing cross product direction (right-hand rule).
• Using degrees instead of radians in differentiation.

Derivative: d(sinθ)/dθ = cosθ, d(cosθ)/dθ = -sinθ

Integral: ∫ sinθ dθ = -cosθ + C

Vector addition magnitude: √(A²+B²+2ABcosθ)

Dot product: A·B = ABcosθ

Cross product magnitude: ABsinθ

Area of parallelogram: |A×B|