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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.
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.
The values of (), (), and are represented by the ordinates of points A, B, and D, respectively, while the values of (), (), and () are represented by the abscissas of points A, C and E, respectively.
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.
Thus, for example, / = is a constructible angle because 15 is the product of the Fermat primes 3 and 5. Similarly π / 12 = 15 ∘ {\displaystyle \pi /12=15^{\circ }} is a constructible angle because 12 is a power of two (4) times a Fermat prime (3).
The concept of unit circle (the set of all vectors of norm 1) is different in different norms: for the 1-norm, the unit circle is a square oriented as a diamond; for the 2-norm (Euclidean norm), it is the well-known unit circle; while for the infinity norm, it is an axis-aligned square.
English: A unit circle with sine (sin), cosine (cos), tangent (tan), cotangent (cot), versine (versin), coversine (cvs), exsecant (exsec), excosecant (excsc) and (indirectly) also secant (sec), cosecant (csc) as well as chord (crd) and arc labeled as trigonometric functions of angle theta. It is designed as alternative construction to "Circle ...