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}.


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

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