Month: August 2023

First Converse, Then Inverse!

This post dedicated to the memory of Professor Tom M. Apostol on the occasion of the 100th anniversary of his birthday. For more details, see my paper here!

Converse of Elementary Functions and Their Fundamental Formulas:

First Formula of Radical:


\boxed{(\sqrt x)^2=x}


Second Formula of Radical:


\boxed{\sqrt{x^2}=\begin{cases}x,~x\ge0\\-x,~x<0\end{cases}}


First Formula of Cube Root Radical:


\boxed{(\sqrt[3]x)^3=x}


Second Formula of Cube Root Radical:


\boxed{\sqrt[3]{x^3}=x}


First Formula of Logarithm:


\boxed{2^{\log_2(x)}=x}


Second Formula of Logarithm:


\boxed{\log_2(2^x)=x}


First Formula of Arc Sine:


\boxed{\sin(\text{Arcsin}(x))=x}


Second Formula of Arc Sine:


\boxed{\text{Arcsin}(\sin(x))=\begin{cases}x,~\text{first quadrant}\\\pi-x,~\text{second and third quadrants}\\x-2\pi,~\text{fourth quadrant}\end{cases}}


First Formula of Arc Cosine:


\boxed{\cos(\text{Arccos}(x))=x}


Second Formula of Arc Cosine:


\boxed{\text{Arccos}(\cos(x))=\begin{cases}x,~\text{first and second quadrants}\\2\pi-x,~\text{third and fourth quadrants}\end{cases}}


First Formula of Arc Tangent:


\boxed{\tan(\text{Arctan}(x))=x}


Second Formula of Arc Tangent:


\boxed{\text{Arctan}(\tan(x))=\begin{cases}x,~\text{first quadrant}\\x-\pi,~\text{second and third quadrants}\\x-2\pi,~\text{fourth quadrant}\end{cases}}