Key Formulas and Theorems for Quick Reference:

Pythagorean Theorem: In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. If ‘a’ and ‘b’ are the lengths of the two shorter sides and ‘c’ is the length of the hypotenuse, then: c^2 = a^2 + b^2.

Quadratic Formula: The solution to a quadratic equation of the form ax^2 + bx + c = 0 is given by: x = (-b ± √(b^2 – 4ac)) / 2a.

Distance Formula: The distance between two points (x₁, y₁) and (x₂, y₂) in a two-dimensional plane is given by: d = √((x₂ – x₁)^2 + (y₂ – y₁)^2).

Slope Formula: The slope ‘m’ of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by: m = (y₂ – y₁) / (x₂ – x₁).

Area of a Triangle: The area ‘A’ of a triangle with base ‘b’ and height ‘h’ is given by: A = 0.5 * b * h.

Area of a Circle: The area ‘A’ of a circle with radius ‘r’ is given by: A = π * r^2.

Circumference of a Circle: The circumference ‘C’ of a circle with radius ‘r’ is given by: C = 2π * r.

Volume of a Cylinder: The volume ‘V’ of a cylinder with base radius ‘r’ and height ‘h’ is given by: V = π * r^2 * h.

Sum of Interior Angles of a Polygon: The sum of the interior angles of a polygon with ‘n’ sides is given by: (n-2) * 180°.

Law of Sines: In any triangle, the ratio of the length of a side to the sine of its opposite angle is the same for all three sides. For a triangle with sides a, b, and c, and angles A, B, and C (opposite sides a, b, and c, respectively): sin(A)/a = sin(B)/b = sin(C)/c.

Law of Cosines: In any triangle, the square of one side is equal to the sum of the squares of the other two sides, minus twice the product of the other two sides and the cosine of the included angle. For a triangle with sides a, b, and c, and angle C (opposite side c): c^2 = a^2 + b^2 – 2ab * cos(C).

Binomial Theorem: For any positive integer ‘n’ and real numbers ‘a’ and ‘b’: (a + b)^n = Σ [nCr * a^(n-r) * b^r], where Σ represents the sum over all values of ‘r’ from 0 to ‘n’, and nCr denotes the binomial coefficient (nCr = n! / (r! * (n-r)!)).

Fundamental Theorem of Calculus: If ‘f(x)’ is a continuous function on the interval [a, b], and ‘F(x)’ is its antiderivative, then the definite integral of ‘f(x)’ from ‘a’ to ‘b’ is equal to F(b) – F(a).

Chain Rule: If ‘y’ is a composite function of ‘u’ and ‘u’ is a function of ‘x’, then the derivative of ‘y’ with respect to ‘x’ (dy/dx) is given by: dy/dx = dy/du * du/dx.

Product Rule: If ‘u(x)’ and ‘v(x)’ are differentiable functions, then the derivative of their product ‘u(x) * v(x)’ with respect to ‘x’ is given by: (u * v)’ = u’ * v + u * v’.

These key formulas and theorems serve as powerful tools for solving a wide range of mathematical problems and understanding fundamental concepts across various mathematical disciplines. They provide a quick reference for students, researchers, and professionals to apply in their mathematical analyses and computations.