Common Formulas and Equations

Mathematics is full of formulas and equations that are used to express relationships between different quantities. Here are some of the most commonly used ones:

Algebra

  • Quadratic Formula: For a quadratic equation ax² + bx + c = 0, the solutions are given by x = [-b ± sqrt(b² - 4ac)] / 2a.
  • Slope-Intercept Form of a Line: y = mx + b, where m is the slope and b is the y-intercept.
  • Distance Formula: The distance between two points (x₁, y₁) and (x₂, y₂) is sqrt[(x₂ - x₁)² + (y₂ - y₁)²].

Geometry

  • Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). So, a² + b² = c².
  • Area of a Triangle: (1/2) * base * height.
  • Circumference of a Circle: 2πr, where r is the radius of the circle.
  • Area of a Circle: πr², where r is the radius of the circle.

Trigonometry

  • Sine, Cosine, and Tangent: For a right triangle, sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, tan(θ) = opposite/adjacent.
  • Law of Sines: For any triangle with sides of length a, b, and c, and opposite angles A, B, and C, sin(A)/a = sin(B)/b = sin(C)/c.
  • Law of Cosines: For any triangle with sides of length a, b, and c, and an angle γ opposite side c, c² = a² + b² - 2ab cos(γ).

Calculus

  • Power Rule for Derivatives: If y = x^n, then dy/dx = n*x^(n-1).
  • Chain Rule for Derivatives: If y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).
  • Fundamental Theorem of Calculus: If F is an antiderivative of f on an interval, then the definite integral of f from a to b is F(b) - F(a).

Remember that these formulas and equations should be used in the appropriate contexts, and you should understand the concepts behind them, not just memorize them.