Flashcard 6 - Next Generation Advanced Algebra and Functions Flashcard Set for the ACCUPLACER Test

The Pythagorean theorem, which relates the side lengths of a right triangle to its hypotenuse, is defined as \(a^2 + b^2 = c^2\), where a and b are the side lengths, and c is the hypotenuse.

The distance formula, defined as \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\), is the Pythagorean formula applied to a coordinate plane.

The trigonometric identity, \(\cos^2\;\theta + \sin^2\;\theta = 1\), is based on the unit circle (with radius 1), where cosine represents the x side length and sine represents the y side length. Similarly, \(1 + \tan^2\;\theta = \sec^2\;\theta\), and \(\cot^2\;\theta + 1 = \csc^2\;\theta\) derive from the same principle.

A thorough understanding of the applications of the Pythagorean theorem, but especially the trigonometric functions and identities, enable many real world scenarios to be modeled and solved.