- Pythagorean Theorem: a^2 + b^2 = c^2
This theorem relates the sides of a right triangle, where 'a' and 'b' are the lengths of the two shorter sides (legs), and 'c' is the length of the hypotenuse. - Quadratic Formula: x = (-b ± sqrt(b^2 - 4ac)) / (2a)
The quadratic formula provides the solutions for a quadratic equation of the form ax^2 + bx + c = 0, where 'a', 'b', and 'c' are coefficients and 'x' represents the unknown variable. - Fundamental Theorem of Calculus: ∫(a to b) f(x) dx = F(b) - F(a)
This theorem connects the concept of differentiation and integration, stating that the definite integral of a function 'f(x)' over an interval [a, b] can be evaluated by finding the antiderivative 'F(x)' of 'f(x)' and subtracting the values at the endpoints. - Euler's Formula: e^(iπ) + 1 = 0
Euler's formula establishes a connection between the exponential function, complex numbers, and trigonometry. It combines five fundamental mathematical constants: e (Euler's number), i (the imaginary unit), π (pi), 0 (zero), and 1 (one). - Law of Cosines: c^2 = a^2 + b^2 - 2ab * cos(C)
The Law of Cosines relates the lengths of the sides of any triangle to the cosine of one of its angles. 'a', 'b', and 'c' are the lengths of the sides, and 'C' is the angle opposite the side 'c'. - Fundamental Identity of Trigonometry: sin^2(x) + cos^2(x) = 1
This identity is a fundamental relationship in trigonometry, stating that the square of the sine of an angle plus the square of the cosine of the same angle is always equal to 1. - Fundamental Theorem of Algebra: Every polynomial equation of degree n has exactly n complex roots, counting multiplicities.
The fundamental theorem of algebra states that every polynomial equation with complex coefficients has at least one complex root. - Binomial Theorem: (a + b)^n = C(n, 0) * a^n * b^0 + C(n, 1) * a^(n-1) * b^1 + ... + C(n, n) * a^0 * b^n
The binomial theorem provides a formula for expanding a binomial raised to a positive integer power 'n'. It involves the binomial coefficients, represented by C(n, k). - Law of Sines: sin(A) / a = sin(B) / b = sin(C) / c
The Law of Sines relates the ratios of the lengths of the sides of a triangle to the sines of the opposite angles. 'a', 'b', and 'c' are the side lengths, and 'A', 'B', and 'C' are the corresponding angles. - Fundamental Property of Logarithms: log(base a)(xy) = log(base a)(x) + log(base a)(y)
This property of logarithms states that the logarithm of a product of two numbers is equal to the sum of the logarithms of the individual numbers, with the same base 'a'.