Trigonometry

Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies.

The 3rd-century astronomers first noted that the lengths of the sides of a right-angle triangle and the angles between those sides have fixed relationships: that is, if at least the length of one side and the value of one angle is known, then all other angles and lengths can be determined algorithmically. These calculations soon came to be defined as the trigonometric functions and today are pervasive in both pure and applied mathematics: fundamental methods of analysis such as the Fourier transform, for example, or the wave equation, use trigonometric functions to understand cyclical phenomena across many applications in fields as diverse as physics, mechanical and electrical engineering, music and acoustics, astronomy, ecology, and biology. Trigonometry is also the foundation of surveying.

Khan Academy

Khan Academy is a non-profit educational organization created in 2006 by educator Salman Khan with the aim of providing a free, world-class education for anyone, anywhere. The organization produces short lectures in the form of YouTube videos. In addition to micro lectures, the organization's website features practice exercises and tools for educators. All resources are available for free to anyone around the world. The main language of the website is English, but the content is also available in other languages.

History

The founder of the organization, Salman Khan, was born in New Orleans, Louisiana, United States to immigrant parents from Bangladesh and India. After earning three degrees from the Massachusetts Institute of Technology (a BS in mathematics, a BS in electrical engineering and computer science, and an MEng in electrical engineering and computer science), he pursued an MBA from Harvard Business School.

In late 2004, Khan began tutoring his cousin Nadia who needed help with math using Yahoo!'s Doodle notepad.When other relatives and friends sought similar help, he decided that it would be more practical to distribute the tutorials on YouTube. The videos' popularity and the testimonials of appreciative students prompted Khan to quit his job in finance as a hedge fund analyst at Connective Capital Management in 2009, and focus on the tutorials (then released under the moniker "Khan Academy") full-time.

Unit circle

In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, the unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. The unit circle is often denoted S¹; the generalization to higher dimensions is the unit sphere.

If (x, y) is a point on the unit circle's circumference, then |x| and |y| are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equation

Since x² = (−x)² for all x, and since the reflection of any point on the unit circle about the x- or y-axis is also on the unit circle, the above equation holds for all points (x, y) on the unit circle, not only those in the first quadrant.

The interior of the unit circle is called the open unit disk, while the interior of the unit circle combined with the unit circle itself is called the closed unit disk.

One may also use other notions of "distance" to define other "unit circles", such as the Riemannian circle; see the article on mathematical norms for additional examples.