Decode Mathematics

 Introduction - Trigonometry is a branch of mathematics which studies about the relationship between the angles and side lengths of the triangles. The “Trigono” word means triangle and the “metry” word means to measure. It has various applications in navigation, astronomy, optics and so on.   They say Trigonometry is all about triangles, but in reality it’s about circles .   The following will help you visualize the different Trigonometric functions like Sine, Cosine, Tangent, Secant, Cosecant and Cotangent.   Visualizing Sine & cosine – In a right angled triangle:  Sine of an acute angle is defined as the ratio of length of the opposite side to its hypotenuse. And  Cosine of an acute angle is defined as the ratio of length of the adjacent side to its hypotenuse. If the length of the hypotenuse is 1 unit, then the Unit circle formed by the hypotenuse will have - `\color{orange} {\sin\theta = \frac{opposite}{hypoten use}}`  `\color{bl...

  The first thing that we must know is that integration and differentiation are reverse process of each other. So, every rule in integration is somewhere related to the rules in differentiation.   The reason, why substitution works is due to the chain rule in differentiation. The following will build an intuition   on how   chain rule connects to the substitution. For indefinite integration. ( note :-  using  substitution  to evaluate a  definite   integral  requires a change to the limits of  integration )   Suppose we have two continuous functions f(x) and g(x),the derivative of f[g(x)] is as follows   `frac{\text{d}f[g(x)]}{\text{d}x}= f'[g(x)].g'(x)` If we integrate both sides with respect to x we get -   f[g(x)] + c =  `\int_{}^{} f'[g(x)].g'(x)` (RHS)                      (LHS)   Le...