Formulae: sinh-1 x = loge (x + √(x² + 1)) cosh-1 x = loge (x + √(x² – 1)) for x ≥ 1 tanh-1 x = ½ log [(1 +x)/ (1 – x)] for x ϵ (-1, 1) coth-1 x = ½ log [(1 +x)/ (1 – x)] for |x| > 1 sech-1 x = Read more about Inverse Hyperbolic functions[…]
Angle: Consider a ray. If this ray rotates about its end point O and takes the position OB, then we say the angle has been generated.Angle is considered as the figure obtained by rotating a given ray about its end point. Measure of an angle is the amount of rotation from initial side to terminal side. System Read more about Trigonometric Ratios, Domains and Range[…]
formulae: sin (A + B) + sin (A – B) = 2 sinA cosB sin (A + B) – sin (A – B) = 2 cosA sinB cos (A + B) + cos (A – B) = 2 cosA cosB cos (A – B) – cos (A + B) = 2 sinA sinB sinC + Read more about Transformation formulae[…]
Property VIII: (i) . (ii) . Property IX: (i) . (ii) . Property X: (i) . (ii) . Property XI: (i) . (ii) . Property XII: (i) . (ii) . Property XIII: (i) sin–¹ x = cos–¹ √(1 – x²) = tan–¹ [x / √(1 – x²)] = cot–¹ √(1 – x²) / x = Read more about Properties of inverse trigonometric functions II[…]
Property I: sin¯¹ (sin θ) = θ for all θ ϵ [-π / 2, π / 2] cos¯¹ (cos θ) = θ for all θ ϵ [0, π] tan¯¹ (tan θ) = θ for all θ Read more about Properties of inverse trigonometric functions I[…]
Consider a system of simultaneous linear equations given by … (1) A set of values of the variables x, y, z which simultaneously satisfy these three equations is called a solution. A system of linear equations has a solution (whether unique or not) the system is said to be consistent. If it has no solution Read more about Solution of simultaneous linear equations[…]
Property 1: Let A = [aij] be a square matrix of order then the sum of the product of elements of any row (columns) with their cofactors is always equal to |A| or det (A) i.e, and Property 2: Let A = [aij] be a square matrix of order n, then the sum of the Read more about PROPERTIES OF DETERMINANTS[…]
Let and be two determinants. Then, the product Δ1Δ2 is defined as = This is row-by-column multiplication value for finding the product of two determinants and it is same as the rule of multiplication of two matrices. Since the value of a determinant does not alter by interchanging the rows and columns. So, we can Read more about Product of determinants[…]
Minor: Let A = [aij] be a square matrix of order n. Then the minor Mij of aij in A is the determinant of the square sub-matrix of order (n – 1) obtained by leaving ith row and jth column of A. Let . M₁₁ = Minor of a₁₁ = . M₁₂ = Minor of Read more about Minor and Co-factors of determinants[…]
Limit of a determinant: Let then provided each of nine limiting values exist finitely. Differentiation of a determinant: Let then . Integration of a determinant: Let where a, b, c, l, m and n are constants then . If more than one row (column) of Δ (x) are variable, then in order to find first we evaluate the determinant Δ (x) Read more about Limit, Differentiation and Integration of Determinant[…]