Definite Integration of Odd and Even Functions – Property II If f(x) is an odd function, then , is an even function. Proof: , , Let t = -y dt = – dy , , As given f is an odd function , = 0 , , Hence, , is an even function. Example: Evaluate Read more about Definite Integration of Odd and Even Functions – Property II[…]
Definite Integration of Odd and Even Functions – Property 1 Property 1: Proof: Put x = -t in first term on R.H.S. Differentiation with respect to ‘x’ dx = – dt when x = -a x = -t -a = -t a = t and x = -t put x = 0 0 = -t Read more about Definite Integration of Odd and Even Functions – Property 1[…]
Evaluation of Trigonometric Limits – Tan , Proof: , , , , , , , Hence proved Example: Evaluate , Solution: , , , , Let us consider , , , = a/2 Share Tweet View Email Print Follow
Evaluation of Trigonometric Limits (where θ is in radius) Proof: Consider a circle of radius r. let O be the center of the circle such that , where θ is measured in radians and its value is very small. Suppose the tangent at A meets OB produced at P. From fig we have Area of Read more about Evaluation of Trigonometric Limits[…]
Solution of Triangles – Part 1 The three sides a, b, c and the three angles A, B, C are called the elements of the triangle ABC, when any three of these six elements(except all three angles) of a triangle are given, the triangle is know as completely that is the other three elements and Read more about Solution of Triangles – Part 1[…]
Slope (Gradient) of a Line Slope of a Line: A line in a coordinate plane form two angles with the x – axis which are supplementary. The angle (say) θ made by the line with the positive direction of the x – axis and measured anticlockwise is called the inclination of the line. The trigonometric Read more about Slope (Gradient) of a Line[…]
Angle between the Pair of Lines An angle between the Pair of Line represented by ax² + 2hxy + by² = 0: Let θ be the angle between the lines. Then, , , , ax² + 2hxy + by² = 0 Sum of the roots is = -2h/b m₁ + m₂ = -2h/b Product of Read more about Angle between the Pair of Lines[…]
Intercepts of a Line If a line cuts x – axis at A (a, 0) and y – axis at B (0, b), then a is called x – intercepts and b is called y – intercepts of the line. Examples: If a line cuts x – axis at (2, 0), then x – intercept Read more about Intercepts of a Line[…]
Menelaw’s Theorem Theorem: If a transversal cut the sides BC, CA, AB of a triangle in D, E, F respectively then BD/DC. CE/EA. AF/FB = -1. Proof: Let A (x₁, y₁), B (x₂, y₂) and C (x₃, y₃) be Let the transversal be ax + by + c = 0 BD/DC = The ratio in Read more about Menelaw’s Theorem[…]
Ceva’s Theorem Theorem: If the lines joining any point P, to the vertices A, B, C of a triangle meet the opposite Sides in D, E, F respectively then (BD.CE.AF)/(DC.EA.FB)=1 Proof: Let A (x₁, y₁), B (x₂, y₂), and C (x₃, y₃) be the vertices. Let us consider point P is (0, 0) We know Read more about Ceva’s Theorem[…]