# Trigonometry

## Trigonometry Topics

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### Subtend

In the triangle illustrated above, side subtends angle . More generally, given a geometric object in the plane and a point , let be the angle from one edge of to the other with vertex at . Then is said to subtend an angle from .

### Full angle

A full angle, also called a complete angle, round angle, or perigon, is an angle equal to radians corresponding to the central angle of an entire circle.Four right angles or two straightangles equal one full angle.

### Angle standard position

An angle drawn on the coordinate plane is said to be in standard position if its initial side lies on the positive x-axis so that its vertex coincides with the origin and its rotation is in the counterclockwise direction.In the above image, the angle is in standard position due to the locations of its vertex and its initial side and because of the direction of its rotation.

### Right angle

A right angle is an angle equal to half the angle from one end of a line segment to the other. A right angle is radians or . A triangle containing a right angle is called a right triangle. However, a triangle cannot contain more than one right angle, since the sum of the two right angles plus the third angle would exceed the total possessed by a triangle.The patterns of cracks observed in mud that has been dried by the sun form curves that often intersect in right angles (Williams 1979, p. 45; Steinhaus 1999, p. 88; Pearce 1990, p. 12).

### Exterior angle bisector

The exterior angle bisectors (Johnson 1929, p. 149), also called the external angle bisectors (Kimberling 1998, pp. 18-19), of a triangle are the lines bisecting the angles formed by the sides of the triangles and their extensions, as illustrated above.Note that the exterior angle bisectors therefore bisect the supplementaryangles of the interior angles, not the entire exterior angles.There are therefore three pairs of oppositely oriented exterior angle bisectors. The exterior angle bisectors intersect pairwise in the so-called excenters , , and . These are the centers of the excircles, i.e., the three circles that are externally tangent to the sides of the triangle (or their extensions).The points determined on opposite sides of a triangle by an angle bisector from each vertex lie on a straight line if either (1) all or (2) one out of the three bisectors is an external angle bisector (Johnson 1929, p. 149; Honsberger..

### Exterior angle

An exterior angle of a polygon is the angle formed externally between two adjacent sides. It is therefore equal to , where is the corresponding internal angle between two adjacent sides (Zwillinger 1995, p. 270).Consider the angles formed between a side of a polygon and the extension of an adjacent side. Since there are two directions in which a side can be extended, there are two such angles at each vertex. However, since corresponding angles are opposite, they are also equal.Confusingly, a bisector of an angle is known as an exterior angle bisector, while a bisector of an angle (which is simply a line oriented in the opposite direction as the interior angle bisector) is not given any special name.The sum of the angles in a convex polygon is equal to radians (), since this corresponds to one complete rotation of the polygon...

### Angle bisector

The (interior) bisector of an angle, also called the internal angle bisector (Kimberling 1998, pp. 11-12), is the line or line segment that divides the angle into two equal parts.The angle bisectors meet at the incenter , which has trilinear coordinates 1:1:1.The length of the bisector of angle in the above triangle is given bywhere and .The points , , and have trilinear coordinates , , and , respectively, and form the vertices of the incentral triangle.

### Direction cosine

Let be the angle between and , the angle between and , and the angle between and . Then the direction cosines are equivalent to the coordinates of a unit vector ,(1)(2)(3)From these definitions, it follows that(4)To find the Jacobian when performing integrals overdirection cosines, use(5)(6)(7)The Jacobian is(8)Using(9)(10)(11)(12)so(13)(14)(15)(16)Direction cosines can also be defined between two sets of Cartesiancoordinates,(17)(18)(19)(20)(21)(22)(23)(24)(25)Projections of the unprimed coordinates onto the primed coordinates yield(26)(27)(28)(29)(30)(31)and(32)(33)(34)(35)(36)(37)Projections of the primed coordinates onto the unprimed coordinates yield(38)(39)(40)(41)(42)(43)and(44)(45)(46)Using the orthogonality of the coordinate system, it must be true that(47)(48)giving the identities(49)for and , and(50)for . These two identities may be combined into the single identity(51)where is the..

### Angle

Given two intersecting lines or line segments, the amount of rotation about the point of intersection (the vertex) required to bring one into correspondence with the other is called the angle between them. The term "plane angle" is sometimes used to distinguish angles in a plane from solid angles measured in space (International Standards Organization 1982, p. 5).The term "angle" can also be applied to the rotational offset between intersecting planes about their common line of intersection, in which case the angle is called the dihedral angle of the planes.Angles are usually measured in degrees (denoted ), radians (denoted rad, or without a unit), or sometimes gradians (denoted grad).The concept of an angle can be generalized from the circle to the sphere, in which case it is known as solid angle. The fraction of a sphere subtended by an object (its solid angle) is measured in steradians, with the entire sphere..

The radian is a unit of angular measure defined such that an angle of one radian subtended from the center of a unit circle produces an arc with arc length 1.A full angle is therefore radians, so there are per radians, equal to or 57./radian. Similarly, a right angle is radians and a straight angle is radians.Radians are the most useful angular measure in calculus because they allow derivative and integral identities to be written in simple terms, e.g.,for measured in radians.Unless stated otherwise, all angular quantities considered in this work are assumed to be specified in radians.

### Solid angle

The solid angle subtended by a surface is defined as the surface area of a unit sphere covered by the surface's projection onto the sphere. This can be written as(1)where is a unit vector from the origin, is the differential area of a surface patch, and is the distance from the origin to the patch. Written in spherical coordinates with the colatitude (polar angle) and for the longitude (azimuth), this becomes(2)Solid angle is measured in steradians, and the solid angle corresponding to all of space being subtended is steradians.To see how the solid angle of simple geometric shapes can be computed explicitly, consider the solid angle subtended by one face of a cube of side length centered at the origin. Since the cube is symmetrical and has six sides, one side obviously subtends steradians. To compute this explicitly, rewrite (1) in Cartesian coordinates using(3)(4)and(5)(6)Considering the top face of the cube, which is located at and has sides..

### Degree

The word "degree" has many meanings in mathematics.The most common meaning is the unit of angle measure defined such that an entire rotation is . This unit harks back to the Babylonians, who used a base 60 number system. likely arises from the Babylonian year, which was composed of 360 days (12 months of 30 days each). The degree is subdivided into 60 arc minutes per degree, and 60 arc seconds per arc minute. In the Wolfram Language, the symbol giving the number of radians in one degree is Degree.The word "degree" is also used in many contexts where it is synonymous with "order," as applied for example to polynomials.

### Trigonometry angles--pi/15

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)These can be derived using the trigonometricaddition formulas(25)(26)(27)(28)and(29)(30)(31)(32)

### Trigonometry angles

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..

Trigonometric functions of for an integer cannot be expressed in terms of sums, products, and finite root extractions on real rational numbers because 13 is not a Fermat prime. This also means that the tridecagon is not a constructible polygon.However, exact expressions involving roots of complex numbers can still bederived using the multiple-angle formula(1)where is a Chebyshev polynomial of the first kind. Plugging in gives(2)Letting and then gives(3)But this is a sextic equation has a cyclic Galois group, and so , and hence , can be expressed in terms of radicals (of complex numbers). The explicit expression is quite complicated, but can be generated in the Wolfram Language using Developer$TrigToRadicals[Sin[Pi/13]].The trigonometric functions of can be given explicitly as the polynomial roots(4)(5)(6)(7)(8)(9)From one of the Newton-Girard formulas,(10)The trigonometric functions of also obey the identities(11)(12)(P. Rolli,.. ### Werner formulas The Werner formulas are the trigonometric product formulas(1)(2)(3)(4)This form of trigonometric functions can be obtained in the WolframLanguage using the command TrigReduce[expr]. ### Trigonometric power formulas Power formulas include(1)(2)(3)and(4)(5)(6)(Beyer 1987, p. 140). Formulas of these types can also be given analytically as(7)(8)(9)(10)where is a binomial coefficient.Additional useful power identities include(11)which the Wolfram Language's FullSimplify command unfortunately does not know about. ### Trigonometry angles--pi/11 Trigonometric functions of for an integer cannot be expressed in terms of sums, products, and finite root extractions on real rational numbers because 11 is not a Fermat prime. This also means that the hendecagon is not a constructible polygon.However, exact expressions involving roots of complex numbers can still bederived using the multiple-angle formula(1)where is a Chebyshev polynomial of the first kind. Plugging in gives(2)Letting and then gives(3)But this quintic equation has a cyclic Galois group, and so , and hence , can be expressed in terms of radicals (of complex numbers). The explicit expression is quite complicated, but can be generated in the Wolfram Language using Developer$TrigToRadicals[Sin[Pi/11]].The trigonometric functions of can be given explicitly as the polynomial roots(4)(5)(6)(7)(8)(9)From one of the Newton-Girard formulas,(10)(11)(12)The trigonometric functions of also obey the identity(13)..

Angle addition formulas express trigonometric functions of sums of angles in terms of functions of and . The fundamental formulas of angle addition in trigonometry are given by(1)(2)(3)(4)(5)(6)The first four of these are known as the prosthaphaeresisformulas, or sometimes as Simpson's formulas.The sine and cosine angle addition identities can be compactly summarized by thematrix equation(7)These formulas can be simply derived using complex exponentials and the Euler formula as follows.(8)(9)(10)(11)Equating real and imaginary parts then gives (1) and (3), and (2) and (4) follow immediately by substituting for .Taking the ratio of (1) and (3) gives the tangent angle addition formula(12)(13)(14)(15)The double-angle formulas are(16)(17)(18)(19)(20)Multiple-angle formulas are given by(21)(22)and can also be written using the recurrence relations(23)(24)(25)The angle addition formulas can also be derived purely algebraically..

### Trigonometry angles--pi/10

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)To derive these formulas, use the half-angle formulas(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)

### Wallis formula

The Wallis formula follows from the infinite productrepresentation of the sine(1)Taking gives(2)so(3)(4)(OEIS A052928 and A063196).An accelerated product is given by(5)(6)where(7)(Guillera and Sondow 2005, Sondow 2005). This is analogous to the products(8)and(9)(Sondow 2005).A derivation of equation (◇) due to Y. L. Yung (pers. comm., 1996; modified by J. Sondow, pers. comm., 2002) defines(10)(11)(12)where is a polylogarithm and is the Riemann zeta function, which converges for . Taking the derivative of (11) gives(13)which also converges for , and plugging in then gives(14)(15)(16)Now, taking the derivative of the zeta function expression (◇) gives(17)and again setting yields(18)(19)(20)(21)where(22)(OEIS A075700) follows from the Hadamard product for the Riemann zeta function. Equating and squaring (◇) and (◇) then gives the Wallis formula.This derivation of the..

### Trigonometry angles--pi/9

Trigonometric functions of radians for an integer not divisible by 3 (e.g., and ) cannot be expressed in terms of sums, products, and finite root extractions on rational numbers because 9 is not a product of distinct Fermat primes. This also means that the regular nonagon is not a constructible polygon.However, exact expressions involving roots of complex numbers can still bederived using the trigonometric identity(1)Let and . Then the above identity gives the cubic equation(2)(3)This cubic is of the form(4)where(5)(6)The polynomial discriminant is then(7)There are therefore three real distinct roots, which are approximately , 0.3420, and 0.6428. We want the one in the first quadrant, which is approximately 0.3420.(8)(9)(10)(11)Similarly,(12)(13)Because of the Vieta's formulas, we have the identities(14)(15)(16)(15) is known as Morrie's law.Ramanujan found the interesting identity(17)(Borwein and Bailey 2003, p. 77;..

### Trigonometry angles--pi/8

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)To derive these formulas, use the half-angle formulas(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)

### Prosthaphaeresis formulas

The Prosthaphaeresis formulas, also known as Simpson's formulas, are trigonometry formulas that convert a product of functions into a sum or difference. They are given by(1)(2)(3)(4)This form of trigonometric functions can be obtained in the WolframLanguage using the command TrigFactor[expr].These can be derived using the above figure (Kung 1996). From the figure, define(5)(6)Then we have the identity(7)(8)(9)(10)Trigonometric product formulas for the difference of the cosines and sines of two angles can be derived using the similar figure illustrated above (Kung 1996). With and as previously defined, the above figure gives(11)(12)(13)(14)

### Newton's formulas

Let a triangle have side lengths , , and with opposite angles , , and . Then(1)(2)(3)

### Trigonometry angles--pi/6

Construction of the angle produces a 30-60-90 triangle, which has angles and . From the above diagram, write for the vertical leg, then the horizontal leg is given by(1)by the Pythagorean theorem. Now use the double-angle formula(2)to obtain(3)which can be solved for to yield(4)Filling in the rest of the trigonometric functions then gives(5)(6)(7)(8)(9)(10)

### Trigonometry angles--pi/5

The trigonometric formulas for can be derived using the multiple-angle formula(1)Letting and then gives(2)Factoring out one power of gives(3)Solving the quadratic equation for gives(4)But must be less than(5)so taking the minus sign and simplifying gives(6)Filling in the remainder of the trigonometry functions then gives(7)(8)(9)(10)(11)(12)and(13)(14)(15)(16)(17)(18)

### Wallis cosine formula

(1)(2)(3)where is a gamma function and is a double factorial.

### Polar angle

There are two different definitions of the polar angle.In the plane, the polar angle is the counterclockwise angle from the x-axis at which a point in the -plane lies.In spherical coordinates, the polar angle is the angle measured from the -axis, denoted in this work, and also variously known as the zenith angle and colatitude.

### Tomahawk

A geometric implement discovered in a 19th century book, and whose inventor is unknown. It essentially consists of a semicircle, a segment which prolongs its diameter and is equal to the radius, and a segment perpendicular to it.It can be used to trisect an angle, an operation impossible with straightedge and compass. If it is adjusted to an the angle so that 1. The line passes through . 2. lies on line . 3. The line is tangent to the semicircle. Then angle is equal to one third of .The interesting fact about the tomahawk is that it can be easily constructed with straightedge and compass. Hence these tools are, from a merely practical point of view, sufficient to trisect an angle. This does not contradict what is known from mathematical theory, since the procedure of shifting a figure on the paper until its parts fall in given positions is not an Euclidean construction...

### Angle trisection

Angle trisection is the division of an arbitrary angle into three equal angles. It was one of the three geometric problems of antiquity for which solutions using only compass and straightedge were sought. The problem was algebraically proved impossible by Wantzel (1836).Although trisection is not possible for a general angle using a Greek construction, there are some specific angles, such as and radians ( and , respectively), which can be trisected. Furthermore, some angles are geometrically trisectable, but cannot be constructed in the first place, such as (Honsberger 1991). In addition, trisection of an arbitrary angle can be accomplished using a marked ruler (a Neusis construction) as illustrated above (Courant and Robbins 1996).An angle can also be divided into three (or any whole number) of equal parts using the quadratrix of Hippias or trisectrix.An approximate trisection is described by Steinhaus (Wazewski 1945; Peterson 1983;..

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