Proof of the Sum of Square Numbers

In the first part of this series, we have counted  the number of squares on a chessboard, and we have discovered that it is equal to the sum of the squares of the first 8 positive integers. The numbers $1^2$, $2^2$, $3^2$ and so on are called  square numbers.

This method can be generalized to compute for the number of squares on larger square boards. If the measure of a board is $n \times n$, then the number of squares on it is » Read more

The Proof of the Tangent Half-Angle Formula

In this post, we prove the following trigonometric identity:

$\displaystyle \tan \frac{\theta}{2} = \frac{\sin\theta}{1 + \cos \theta} = \frac{1 - \cos \theta}{\sin \theta}$.

Proof

Consider a semi-circle with “center” $O$ and diameter $AB$ and radius equal to 1 unit as shown below.  If we let $\angle BOC =\theta$, then by the Inscribed Angle Theorem, $\angle CAB = \frac{\theta}{2}$.

Draw $CD$ perpendicular to $OB$ as shown in the second figure. We can compute for the sine and cosine of $\theta$ which equal to the lengths of $CD$ and $OD$, respectively. In effect, $BD = 1 - \cos \theta$ and $AD = 1 + \cos \theta$. » Read more

Mathematics and Multimedia Blog Carnival #3

Welcome the third edition of the Mathematics and Multimedia blog Carnival. This will be the last edition that I will be hosting this year. The fourth edition  will be hosted by Wild About Math!.

Before we begin, let’s have some interesting trivia about the number three. » Read more

1 2