Distance between two points

In analytical geometry, the distance between two points in a plane is given by the distance formula: d = sqrt((x2 – x1)^2 + (y2 – y1)^2) where (x1, y1) and (x2, y2) are the coordinates of the two points and d is the distance between them. To use the formula, simply substitute the values of…

Inverse trigonometric functions (principal value only) and their elementary properties

Inverse trigonometric functions are functions that return the angle whose trigonometric ratio is a given value. Here are the principal value only of inverse trigonometric functions and their elementary properties: It is important to note that inverse trigonometric functions are not always unique, as they depend on the quadrant in which the angle lies. For…

Formulae involving multiple and sub-multiple angles

Here are some of the commonly used trigonometric formulas involving multiple and sub-multiple angles: These formulas can be used to simplify trigonometric expressions, solve equations, and prove identities. What is Required Formulae involving multiple and sub-multiple angles Here are some of the commonly used required trigonometric formulas involving multiple and sub-multiple angles: These formulas can…

Trigonometric functions

Trigonometric functions are mathematical functions that relate the angles of a right triangle to the lengths of its sides. The most commonly used trigonometric functions are sine (sin), cosine (cos), and tangent (tan). These functions can be defined in terms of the sides of a right triangle as follows: In addition to these three functions,…

Standard deviation and variance of grouped and ungrouped data

Standard deviation and variance are measures of variability that provide information about how spread out a dataset is. The standard deviation is the square root of the variance, and it measures the average deviation of each data point from the mean. In general, the larger the standard deviation or variance, the more spread out the…

Mean deviation

Mean deviation is a measure of variability that describes the average difference between the values in a dataset and their mean. It is also known as mean absolute deviation (MAD). The formula for calculating the mean deviation is: Mean Deviation = (Σ |xi – x̄|) / n where: Σ represents the sum of the absolute…

Determinant of a square matrix of order up to three

The determinant of a square matrix of order up to three can be calculated as follows: For a 1×1 matrix: The determinant of a 1×1 matrix is simply the value of the only element in the matrix. For a 2×2 matrix: The determinant of a 2×2 matrix is calculated as follows: |a b| |c d|…

Matrices Elementary row and column transformations

Matrices are mathematical objects consisting of rows and columns of numbers. They are commonly used in many fields of mathematics, physics, engineering, and computer science. One of the most important concepts in matrix theory is the idea of elementary row and column transformations. These are operations that can be performed on a matrix to transform…

Transpose of a matrix

In linear algebra, the transpose of a matrix is an operation that flips the matrix over its diagonal, reflecting its rows and columns. The transpose of a matrix A is denoted by A^T. To compute the transpose of a matrix, you simply write the rows of the matrix as columns, and the columns as rows.…

Multiplication by a scalar and product of matrices

Multiplication by a scalar: Multiplying a matrix by a scalar means multiplying every entry of the matrix by that scalar. For example, if A is a matrix and k is a scalar, then the product kA is obtained by multiplying every entry of A by k. Formally, if A = [a_ij] is an m x…