Basic Mathematics Summary Lecture

Introduction

• Basic Mathematics is crucial for understanding Physics.
• A good Physics question often involves strong mathematical concepts.
• Critical points in problems require checking through differentiation if there are multiple points.
• Concepts like GP series are important for solving various problems.

Key Mathematical Concepts

Trigonometry

• Angle Definition: Angle = Arc length / Radius (Dimensionless), 180° = π radians.
• Right-Angle Triangle: Hypotenuse is opposite 90°, base and perpendicular are relevant.
• Standard Triangles:
  • 3-4-5 Triangle: Angles are 53° and 37°.
  • 5-12-13 Triangle.
• Trigonometric Ratios:
  • Tan θ = Perpendicular / Base.
  • Sin θ = Opposite / Hypotenuse.
  • Cos θ = Adjacent / Hypotenuse.
• Identities:
  • sin²θ + cos²θ = 1.
  • 1 + tan²θ = sec²θ.
  • 1 + cot²θ = csc²θ.
• Angle Transformations: Calculating angles like sin(15°), cos(105°) using known angles and formulas like sin(a ± b).

Quadratic Equations

• General form: ax² + bx + c = 0.
• Discriminant (D) = b² - 4ac:
  • D > 0: Two distinct real roots.
  • D = 0: One real root (coinciding).
  • D < 0: Complex roots.
• Roots: (-b ± √D) / 2a.

Series

• AP (Arithmetic Progression): Common difference between terms.
  • Sum: Sₙ = n/2 × (first term + last term).
• GP (Geometric Progression):
  • Terms: a, ar, ar²,...
  • Sum of infinite GP: a / (1 - r) if |r| < 1.

Equations of Lines and Slopes

• Slope (m): Tan of angle with positive x-axis.
• Equation of a line: y = mx + c.
  • 'm': Slope.
  • 'c': Y-intercept.
• Types of Slopes: Positive, Negative, Zero, Infinite.
• Techniques for finding slope using angle.

Parabola and Hyperbola

• Parabola: y = x², symmetric about y-axis.
• Rectangular Hyperbola: xy = constant.

Circle and Ellipse

• Circle: Special case of ellipse where A = B.
• Ellipse Equation: x²/a² + y²/b² = 1.

Differentiation

• Rate of change of dependent variable with respect to independent variable.
• Basic Rules:
  • Constant differentiation = 0.
  • Power rule: d/dx(xⁿ) = nxⁿ⁻¹.
  • Chain rule for composite functions.
• Product and Quotient Rule.
• Uses in Physics for instantaneous rates.

Integration

• Anti-derivative: Reverse process of differentiation.
• Basic Integrals:
  • ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C.
  • ∫ 1/x dx = ln|x| + C.
  • ∫ eˣ dx = eˣ + C.
• Definite Integrals: Area under the curve.

Maxima and Minima

• Finding Extremes:
  • First derivative test to find critical points.
  • Second derivative test to classify maxima/minima.

Conclusion

• This is a quick revision of topics covered over multiple lectures.
• Understanding these concepts is essential for tackling complex Physics problems.

These notes summarize key concepts in basic mathematics necessary for further studies in physics, focusing on trigonometry, equation solving, series, and differentiation/integration techniques.