The study of angles and of the angular relationships of planar and three-dimensional figures is known as trigonometry. The trigonometric functions (also called the circular functions) comprising trigonometry are the cosecant , cosine , cotangent , secant , sine , and tangent . The inverses of these functions are denoted , , , , , and . Note that the notation here means inverse function, not to the power.The trigonometric functions are most simply defined using the unit circle. Let be an angle measured counterclockwise from the x-axis along an arc of the circle. Then is the horizontal coordinate of the arc endpoint, and is the vertical component. The ratio is defined as . As a result of this definition, the trigonometric functions are periodic with period , so(1)where is an integer and func is a trigonometric function.A right triangle has three sides, which can be uniquely identified as the hypotenuse, adjacent to a given angle , or opposite . A helpful..
The angles (with integers) for which the trigonometric functions may be expressed in terms of finite root extraction of real numbers are limited to values of which are precisely those which produce constructible polygons. Analytic expressions for trigonometric functions with arguments of this form can be obtained using the Wolfram Language function ToRadicals, e.g., ToRadicals[Sin[Pi/17]], for values of (for , the trigonometric functions auto-evaluate in the Wolfram Language).Compass and straightedge constructions dating back to Euclid were capable of inscribing regular polygons of 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 40, 48, 64, ..., sides. However, Gauss showed in 1796 (when he was 19 years old) that a sufficient condition for a regular polygon on sides to be constructible was that be of the form(1)where is a nonnegative integer and the are distinct Fermat primes. Here, a Fermat prime is a prime Fermat number, i.e., a prime number of the..
A 16-sided polygon, sometimes also called a hexakaidecagon. The regular hexadecagon is a constructible polygon, and the inradius , circumradius , and area of the regular hexadecagon of side length 1 are(1)(2)(3)