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The trigonometric functions can be constructed geometrically in terms of a unit circle centered at O. Historically, the versed sine was considered one of the most important trigonometric functions. [12] [37] [38]
Since C = 2πr, the circumference of a unit circle is 2π. In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. [1] Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane.
All of the trigonometric functions of the angle θ (theta) can be constructed geometrically in terms of a unit circle centered at O. Sine function on unit circle (top) and its graph (bottom) In this illustration, the six trigonometric functions of an arbitrary angle θ are represented as Cartesian coordinates of points related to the unit circle.
Trigonometric ratios can also be represented using the unit circle, which is the circle of radius 1 centered at the origin in the plane. [37] In this setting, the terminal side of an angle A placed in standard position will intersect the unit circle in a point (x,y), where x = cos A {\displaystyle x=\cos A} and y = sin A {\displaystyle ...
In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, = = =, where a, b, and c are the lengths of the sides of a triangle, and α, β, and γ are the opposite angles (see figure 2), while R is the radius of the triangle's circumcircle.
A trigonometric number is a number that can be expressed as the sine or cosine of a rational multiple of π radians. [2] Since sin ( x ) = cos ( x − π / 2 ) , {\displaystyle \sin(x)=\cos(x-\pi /2),} the case of a sine can be omitted from this definition.
The haversine formula determines the great-circle distance between two points on a sphere given their longitudes and latitudes.Important in navigation, it is a special case of a more general formula in spherical trigonometry, the law of haversines, that relates the sides and angles of spherical triangles.
The tangent half-angle substitution parametrizes the unit circle centered at (0, 0). Instead of +∞ and −∞, we have only one ∞, at both ends of the real line. That is often appropriate when dealing with rational functions and with trigonometric functions. (This is the one-point compactification of the line.)