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A Farewell to WordPress

To our loyal follows of the Brilliant.org Blog,

We wanted to announce that over the past weeks the publication capabilities of Brilliant.org have been transferred from this wordpress blog to our own main site. Desiring to integrate our blog posts with the problem solving activity done on our weekly challenges, we have transferred our entire blog archives over to our main site. Eventually each blog post will evolve to have sets of practice problems to illustrate the concepts covered in each post. To check it out click here:

See the new home of the Brilliant.org Blog

Whether publishing a news update, an exposition of a key technique, a rumination on the history of math or science, or a brain teaser for fun, we believe our own website can better store host our publications. Our content will now be easier to find and connect better to content that is relevant to it. We hope you like it!

In the next few days, all of our subscribers to this blog will receive an email asking if you would like to continue receiving email updates as new posts are published to the Brilliant.org main site.

Thank you all for considerately following the posts of this blog. Please keep in touch.

All the best,

The staff of Brilliant.org

Trigonometric Identities

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[This post is targeted at a level 2 user. You should have read Trigonometric Functions.]

When students are first exposed to trigonometric identities, they are often given a list of formulas, which they are asked to memorize. Here is a way for you to remember many of these ideas:

Start by drawing a regular hexagon and connect each of the vertices to the center. In the left most vertex, label it \tan . In the bottom left vertex, label it \sin , in the bottom right vertex, label it \cos . In the center, label it 1. Now, to figure out what to label the remaining vertices, simply look at the diagonally opposite vertex and label it as the reciprocal. You should get the following diagram:

Calvin's Trig Hexagon

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Parity II

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348px-Chess_piece_-_Black_knight

The principles of parity can be used to understand the potential chess moves of a Knight.

[This post is targeted at a Level 4 student. You should be familiar with Parity.]

We will further explore the idea of parity, and its various uses. Apart from simply determining if a number is odd or even, parity can be used as a simple counting argument in many cases. In Parity Test Yourself 1, we gave a question where Mary had an even number of sheep at the start, and an odd number of sheep at the end of the day. This implies that she had a different amount of sheep, and hence didn’t do her job properly.

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Probability II

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Four_of_a_Kind_(3263015699)

[This is targeted at a level 3 student. This is a continuation of the Probability blog post.]

In our first blog about probability, we focused on sets of events that are mutually exclusive. In this blog, we will expand to look at events which are not mutually exclusive, and some techniques we can use to solve various types of probability questions.

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Trigonometric functions

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[This is targeted at a Level 1 student.]

The trigonometric functions are functions of an angle, the most prominent of which are sine, cosine and tangent. These are best understood by considering the line segments from a unit circle.

Given an angle \theta , construct the half ray, \ell_\theta , that is an anti-clockwise rotation of \theta from the positive x-axis. The intersection of this ray with the unit circle, labelled as P_\theta is given by P_\theta = ( \cos \theta, \sin \theta ) , which gives the definition of \sin \theta, \cos \theta . We further introduce that \tan \theta = \frac { \sin \theta}{\cos \theta} .

Unit Circle

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Gaussian Integers III

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Gaussian_Integer_Coloring_-_Mandelbrot

A version of the Mandelbrot set colored in relation to the distance of each iterate from the nearest gaussian integer (Image courtesy of wikimedia creative commons).

[This post is targeted at a Level 5 student. This is a continuation of the blog post Gaussian integers II.]

You are probably used to the fact that every positive integer can be uniquely expressed as a product of positive primes, up to the order of multiplication. This property is called the Fundamental Theorem of Arithmetic. While this seems like an obvious fact, can we create a system where it is not true?

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Probability

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2-Dice-Icon.svg

[This post is targeted at a Level 2 student. You should be familiar with the Rule of sum, Rule or product.]

Probability is a measure of how likely it is that a certain event will occur. Probability is a concept that turns up frequently in the real world. It has applications in finance, meteorology, marketing, and many other areas.

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