## How to randomly scatter points inside a circle with ggplot, without clustering around the center?

Solution for How to randomly scatter points inside a circle with ggplot, without clustering around the center? is Given Below: …

## How to compare two tangents?

Solution for How to compare two tangents? is Given Below: How to compare two tangents? The problem is that atan2 …

## In ggplot, how to draw a circle/disk with a line that divides its area according to a given ratio and colored points inside?

Solution for In ggplot, how to draw a circle/disk with a line that divides its area according to a given …

## Analogies between \$(tan, sec)\$ and \$(sinh, cosh)\$

I think everything follows from your first equation. Since the signs of tan, sec go like the signs of sinh, …

## Alternative form of ArcSin[Sin[x]]

We can take the Floor and Ceiling from Carl’s answer and expand them out: PiecewiseExpand[ PowerExpand[ArcSin[Sin[x]], Assumptions -> x ∈ …

## Alternative definition of hyperbolic cosine without relying on exponential function

The more-geometrically-minded of us take \$cosh u\$ and \$sinh u\$ to be defined via the “unit hyperbola”, \$x^2 – y^2 …

## A last year Putnam question maximum \$sum{cos 3x}\$

Visualising the solution You have asked for help in visualising the solution. I think you will find it useful to …

## A curious equation containing an integral \$int_0^{pi/4}arctanleft(tan^xthetaright)dtheta=frac{ln2cdotln x}{16}\$

The only positive answer is \$\$x=3+sqrt{8}.\$\$ The corresponding integral (in a slightly different form) is given as formula (17) and …

## \$Acos(theta) + Bsin(theta)\$ for complex \$A,B\$

The relationship given in ref{22} below (highlighted) is derived as follows: \$\$vec{Z} = vec{A} cos(theta) + vec{B} sin(theta) tag{1} label{1a}\$\$ …