# Browse Classroom Capsules and Notes

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Displaying 981 - 990 of 1212

The author presents a geometric proof of Machin's formula, which expresses $\frac {\pi}{4}$ as the difference of two inverse tangent functions.

The author proves visually the law of cosines.

The author gives two applications of a method for finding a function $g$ such that $f(g(x)) \geq f(x)$.

The author visually expresses the cube of a number as an arithmetic sum.

A visual determination of the area of a right triangle is given using an inscribed circle.

An odd prime $p$ has $(p-1)/2$ quadratic residues mod $p$, and for relatively prime $p$ and $q$ there are $(p-1)(q-1)/2$ non-representable Frobenius numbers. The author discusses a...

An old and forgotten method for determining whether a quartic polynomial over the rationals is reducible is revived.

The Butterfly Theorem is generalized to include Sidney Kung's Butterfly Theorem.

The unique root of $cos(x)=x$, the Dottie number, is a simple non-trivial example of a universal attracting fixed point. The author gives pedagogical examples for several courses ranging from...

A visual solution for an alternating sum of odd squares is presented.