A visual proof that the sine is subadditive on [0, pi].

A visual proof that sin x/x is monotonically increasing on (0, pi/2). For tan x/x, see p. 420 (EJ1017686).

A visual proof that tan x/x is monotonically increasing on (0, pi/2). For sin x/x, see p. 408 (EJ1017684).

Using techniques that show that e and pi are transcendental, we give a short, elementary proof that pi is irrational based on Euler's identity. The proof involves evaluations of a polynomial using repeated applications of Leibniz formula as organized in a Leibniz table.

In my Calculus classes I encourage my students to actively reflect on course material, to work collaboratively, and to generate diverse solutions to questions. To facilitate this I use peer instruction (PI), a structured questioning process, and i-clickers, a radio frequency classroom response system enabling students to vote anonymously. This…

The celebrated Basel Problem, that of finding the infinite sum 1 + 1/ 4 + 1/9 + 1/16 + ..., was open for 91 years. In 1735 Euler showed that the sum is pi[superscript 2]/6. Dozens of other solutions have been found. We give one that is short and elementary.

Using continued fraction expansions, we can approximate constants, such as pi and e, using an appropriate integer n raised to the power x[superscript 1/x], x a suitable rational. We review continued fractions and give an algorithm for producing these approximations.

In 1997, Bailey and Bannister showed that a + b greater than c + h holds for all triangles with [gamma] less than arctan (22/7)where a, b, and c are the sides of the triangle, "h" is the altitude to side "c", and [gamma] is the angle opposite c. In this paper, we show that a + b greater than c + h holds approximately 92% of the time for all…

This paper is a whimsical survey of the various explanations which might account for the biblical passage in I Kings 7:23 that describes a round object--a bronze basin called Solomon's Sea--as having diameter ten cubits and circumference thirty cubits. Can the biblical pi be any number other than 3? We offer seven different perspectives on this…

In this article we consider an example suitable for investigation in many mid and upper level undergraduate mathematics courses. Fourier series provide an excellent example of the differences between uniform and non-uniform convergence. We use Dirichlet's test to investigate the convergence of the Fourier series for a simple periodic saw tooth…

We provide a geometric proof of the formula for the sine of the sum of two positive angles whose measures sum to less than [pi]/2. (Contains 1 figure.)

This article discusses a writing project that offers students the opportunity to solve one of the most famous geometric problems of Greek antiquity; namely, the impossibility of trisecting the angle [pi]/3. Along the way, students study the history of Greek geometry problems as well as the life and achievements of Carl Friedrich Gauss. Included is…

The area of a constant-width band is shown to be twice the length of its middle section times its width "h." The length of the middle section is shown to be equal to the length of the inner boundary plus [pi]"h." These results are obtained using the change of variables formula for double integrals.

A first-year seminar general education course provides a good opportunity to search for mathematical topics associated with the popular culture represented in the course's required films and readings. We discuss mathematical connections to several books, including "Life of Pi" and "The Curious Incident of the Dog in the Night-Time," and to the…

Euler gave a simple method for showing that [zeta](2)=1/1[superscript 2] + 1/2[superscript 2] + 1/3[superscript 2] + ... = [pi][superscript 2]/6. He generalized his method so as to find [zeta](4), [zeta](6), [zeta](8),.... His computations became increasingly more complex as the arguments increased. In this note we show a different generalization…