Functional Equations Functional equation is any equation that specifies a function in implicit form. Often, the equation relates the value of a function (or functions) at some point with its values at other points. For instance, the properties of functions can be determined by considering the types of functional equations the satisfy. Functional equations satisfied Read more about Functional Equations[…]
General and Particular Solution of a Differential Equation We have solved the equation of the type: x² + 1 = 0 … (1) sin²x – cosx = 0 … (2) solution of equation (1) and (2) are numbers, real or complex, that will satisfy the given equation i.e., when that number is substituted for the Read more about General and Particular Solution of a Differential Equation[…]
Range of Expressions using Trigonometric Substitution Range of different trigonometric functions helps us to use these functions as substitution in many algebraic expressions. This simplifies the give expression to another expression, the range of which can be easily determined. Example 1: If x² + y² = 4, then find the maximum value of . Solution: Read more about Range of Expressions using Trigonometric Substitution[…]
Sine of the Difference and Sum of Two Angles 1. sin (A-B) = sinA cosB + cosA sinB Proof: L.H.S: sin (A-B) = sinA cosB + cosA sinB L.H.S sin (A – B) = cos (90 – (A – B)) (since cos (90 – θ) = sin θ) = cos (90 – (A – B)) Read more about Sine of the Difference and Sum of Two Angles[…]
Conditional Identities (Tanθ) – Problems 1. If A + B + C = nπ, n Є Z that TanA + TanB + TanC = TanA TanB TanC. Proof: We have , A + B + C = nπ Tan (A + B + C) = tan(nπ) = tan(π) = 0 , TanA + TanB + Read more about Conditional Identities (Tanθ) – Problems[…]
Conditional Identities Some standard Identities in Triangle: 1. sin2A + sin2B + sin2C = 4 sinA sinB sinC Proof: L.H.S sin2A + sin2B + sin2C (since sinC + sinD = 2sin (C + D)/2 cos (C – D)/2), , , , , , , , , , , , Hence proved 2. cosA + cosB Read more about Conditional Identities[…]
Sum of Sines or Cosines of n Angles in A.P . Proof: Let S = sinα + sin(α + β) + sin(α + 2β) + . . . + sin( α + (n – 1)β) Here angles are in A.P and common difference of angles in β . multiplying both sides by 2sin(β/2), we get Read more about Sum of Sines or Cosines of n Angles in A.P[…]
Trigonometric Identities – Problems Trigonometric identities that involve trigonometric function that are true for every single value of the occurring variable. in other words, they are equation that hold true regardless of the value of the angles being chosen. Trigonometric identities are as follow. sin²θ + cos²θ = 1 sec²θ – tan²θ = 1 cosec²θ Read more about Trigonometric Identities – Problems[…]
Equation of a Plane Bisecting the Angle Between Two Planes Given planes are . . . (1) . . . (2) Let P (x, y, z) be a point on the plane bisecting the angle between (1) and (2). Let PL and PM be the length to the perpendiculars from P to planes (i) and Read more about Equation of a Plane Bisecting the Angle Between Two Planes[…]
Inequalities involving Weighted Means If aᵢ > 0, i = 1, 2, …, n and wᵢ ≥ 0, where i = 1, 2, …, n, then weighted arithmetic mean, , Weighted geometric mean, , Weighted harmonic mean, . From A.M ≥ G.M ≥ H.M. We have Aw ≥ Gw ≥ Hw Example: Prove that . Read more about Inequalities involving Weighted Means[…]