Search results
Results from the WOW.Com Content Network
The six trigonometric functions are defined for every real number, except, for some of them, for angles that differ from 0 by a multiple of the right angle (90°). Referring to the diagram at the right, the six trigonometric functions of θ are, for angles smaller than the right angle:
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.
Visual proof of the Pythagorean identity: for any angle , the point (,) = (, ) lies on the unit circle, which satisfies the equation + =.Thus, + =. In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B (which might contain some variables) produce the same value for all values of the variables ...
In what follows, a capital first letter is used for the principal value of functions, and the lower case version is used for the multivalued function. The single valued version of definitions and identities is always given first, followed by a separate section for the multiple valued versions.
Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics". [2] When x = π, Euler's formula may be rewritten as e iπ + 1 = 0 or e iπ = −1, which is known as Euler's identity.
Euler's identity is also a special case of the more general identity that the n th roots of unity, for n > 1, add up to 0: = = Euler's identity is the case where n = 2. A similar identity also applies to quaternion exponential: let {i, j, k} be the basis quaternions; then,
Get AOL Mail for FREE! Manage your email like never before with travel, photo & document views. Personalize your inbox with themes & tabs. You've Got Mail!
However, there are generalizations of this formula valid for other exponents. These can be used to give explicit expressions for the n th roots of unity, that is, complex numbers z such that z n = 1. Using the standard extensions of the sine and cosine functions to complex numbers, the formula is valid even when x is an arbitrary complex number.