The external secant function (abbreviated exsecant , symbolized exsec ) is a trigonometric function defined in terms of the secant function:
61-574: (Redirected from EXS ) Exs , or EXS may refer to: Exsecant Jet2.com , a British airline Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Exs . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Exs&oldid=1090894940 " Category : Disambiguation pages Hidden categories: Short description
122-407: A circle with one endpoint on the circumference a secans exterior . The trigonometric secant , named by Thomas Fincke (1583), is more specifically based on a line segment with one endpoint at the center of a circle and the other endpoint outside the circle; the circle divides this segment into a radius and an external secant. The external secant segment was used by Galileo Galilei (1632) under
183-1351: A constant η equal to 1 inverse radian (1 rad ) in a fashion similar to the introduction of the constant ε 0 . With this change the formula for the angle subtended at the center of a circle, s = rθ , is modified to become s = ηrθ , and the Taylor series for the sine of an angle θ becomes: Sin θ = sin x = x − x 3 3 ! + x 5 5 ! − x 7 7 ! + ⋯ = η θ − ( η θ ) 3 3 ! + ( η θ ) 5 5 ! − ( η θ ) 7 7 ! + ⋯ , {\displaystyle \operatorname {Sin} \theta =\sin \ x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots =\eta \theta -{\frac {(\eta \theta )^{3}}{3!}}+{\frac {(\eta \theta )^{5}}{5!}}-{\frac {(\eta \theta )^{7}}{7!}}+\cdots ,} where x = η θ = θ / rad {\displaystyle x=\eta \theta =\theta /{\text{rad}}}
244-412: A deviation from a straight line ; the second, angle as quantity, by Carpus of Antioch , who regarded it as the interval or space between the intersecting lines; Euclid adopted the third: angle as a relationship. In mathematical expressions , it is common to use Greek letters ( α , β , γ , θ , φ , . . . ) as variables denoting the size of some angle (the symbol π
305-451: A full turn are not equivalent. To measure an angle θ , a circular arc centered at the vertex of the angle is drawn, e.g., with a pair of compasses . The ratio of the length s of the arc by the radius r of the circle is the number of radians in the angle: θ = s r r a d . {\displaystyle \theta ={\frac {s}{r}}\,\mathrm {rad} .} Conventionally, in mathematics and
366-419: A point on a circle or describing the orientation of an object in two dimensions relative to a reference orientation, angles that differ by an exact multiple of a full turn are effectively equivalent. In other contexts, such as identifying a point on a spiral curve or describing an object's cumulative rotation in two dimensions relative to a reference orientation, angles that differ by a non-zero multiple of
427-414: A straight line, they are supplementary. Therefore, if we assume that the measure of angle A equals x , the measure of angle C would be 180° − x . Similarly, the measure of angle D would be 180° − x . Both angle C and angle D have measures equal to 180° − x and are congruent. Since angle B is supplementary to both angles C and D , either of these angle measures may be used to determine
488-571: A sufficiently small angle, a circular arc is approximately shaped like a parabola , and the versine and exsecant are approximately equal to each-other and both proportional to the square of the arclength. The inverse of the exsecant function, which might be symbolized arcexsec , is well defined if its argument y ≥ 0 {\displaystyle y\geq 0} or y ≤ − 2 {\displaystyle y\leq -2} and can be expressed in terms of other inverse trigonometric functions (using radians for
549-502: A triangle is supplementary to the third because the sum of the internal angles of a triangle is a straight angle. The difference between an angle and a complete angle is termed the explement of the angle or conjugate of an angle. The size of a geometric angle is usually characterized by the magnitude of the smallest rotation that maps one of the rays into the other. Angles of the same size are said to be equal congruent or equal in measure . In some contexts, such as identifying
610-421: A two-dimensional Cartesian coordinate system , an angle is typically defined by its two sides, with its vertex at the origin. The initial side is on the positive x-axis , while the other side or terminal side is defined by the measure from the initial side in radians, degrees, or turns, with positive angles representing rotations toward the positive y-axis and negative angles representing rotations toward
671-443: A wire. In recent years, the availability of calculators and computers has removed the need for trigonometric tables of specialized functions such as this one. Exsecant is generally not directly built into calculators or computing environments (though it has sometimes been included in software libraries ), and calculations in general are much cheaper than in the past, no longer requiring tedious manual labor. Naïvely evaluating
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#1732798325845732-427: Is 1 / 256 of a turn. Plane angle may be defined as θ = s / r , where θ is the magnitude in radians of the subtended angle, s is circular arc length, and r is radius. One radian corresponds to the angle for which s = r , hence 1 radian = 1 m/m = 1. However, rad is only to be used to express angles, not to express ratios of lengths in general. A similar calculation using
793-519: Is log exsec 1° ≈ −3.817 220 , all of whose digits are meaningful. If the logarithm of exsecant is calculated by looking up the secant in a six-place trigonometric table and then subtracting 1 , the difference sec 1° − 1 ≈ 0.000 152 has only 3 significant digits , and after computing the logarithm only three digits are correct, log(sec 1° − 1) ≈ −3.81 8 156 . For even smaller angles loss of precision
854-493: Is "pedagogically unsatisfying". In 1993 the American Association of Physics Teachers Metric Committee specified that the radian should explicitly appear in quantities only when different numerical values would be obtained when other angle measures were used, such as in the quantities of angle measure (rad), angular speed (rad/s), angular acceleration (rad/s ), and torsional stiffness (N⋅m/rad), and not in
915-496: Is clear that the complete form is meant. Current SI can be considered relative to this framework as a natural unit system where the equation η = 1 is assumed to hold, or similarly, 1 rad = 1 . This radian convention allows the omission of η in mathematical formulas. It is frequently helpful to impose a convention that allows positive and negative angular values to represent orientations and/or rotations in opposite directions or "sense" relative to some reference. In
976-402: Is different from Wikidata All article disambiguation pages All disambiguation pages Exsecant exsec θ = sec θ − 1 = 1 cos θ − 1. {\displaystyle \operatorname {exsec} \theta =\sec \theta -1={\frac {1}{\cos \theta }}-1.} It
1037-410: Is in the interior of angle AOC, then m ∠ A O C = m ∠ A O B + m ∠ B O C {\displaystyle m\angle \mathrm {AOC} =m\angle \mathrm {AOB} +m\angle \mathrm {BOC} } I.e., the measure of the angle AOC is the sum of the measure of angle AOB and the measure of angle BOC. Three special angle pairs involve
1098-511: Is independent of the size of the circle: if the length of the radius is changed, then the arc length changes in the same proportion, so the ratio s / r is unaltered. Throughout history, angles have been measured in various units . These are known as angular units , with the most contemporary units being the degree ( ° ), the radian (rad), and the gradian (grad), though many others have been used throughout history . Most units of angular measurement are defined such that one turn (i.e.,
1159-459: Is meant, and in these cases, no ambiguity arises. Otherwise, to avoid ambiguity, specific conventions may be adopted so that, for instance, ∠BAC always refers to the anticlockwise (positive) angle from B to C about A and ∠CAB the anticlockwise (positive) angle from C to B about A. There is some common terminology for angles, whose measure is always non-negative (see § Signed angles ): The names, intervals, and measuring units are shown in
1220-509: Is the natural logarithm . See also Integral of the secant function . The exsecant of twice an angle is: exsec 2 θ = 2 sin 2 θ 1 − 2 sin 2 θ . {\displaystyle \operatorname {exsec} 2\theta ={\frac {2\sin ^{2}\theta }{1-2\sin ^{2}\theta }}.} Van Sickle, Jenna (2011). "The history of one definition: Teaching trigonometry in
1281-462: Is the angle in radians. The capitalized function Sin is the "complete" function that takes an argument with a dimension of angle and is independent of the units expressed, while sin is the traditional function on pure numbers which assumes its argument is a dimensionless number in radians. The capitalised symbol Sin {\displaystyle \operatorname {Sin} } can be denoted sin {\displaystyle \sin } if it
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#17327983258451342-412: Is the angle of the rays lying tangent to the respective curves at their point of intersection. The magnitude of an angle is called an angular measure or simply "angle". Angle of rotation is a measure conventionally defined as the ratio of a circular arc length to its radius , and may be a negative number . In the case of a geometric angle, the arc is centered at the vertex and delimited by
1403-400: Is the figure formed by two rays , called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays are also known as plane angles as they lie in the plane that contains the rays. Angles are also formed by the intersection of two planes; these are called dihedral angles . Two intersecting curves may also define an angle, which
1464-404: Is typically determined by a normal vector passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie. In navigation , bearings or azimuth are measured relative to north. By convention, viewed from above, bearing angles are positive clockwise, so a bearing of 45° corresponds to a north-east orientation. Negative bearings are not used in navigation, so
1525-447: Is typically not used for this purpose to avoid confusion with the constant denoted by that symbol ). Lower case Roman letters ( a , b , c , . . . ) are also used. In contexts where this is not confusing, an angle may be denoted by the upper case Roman letter denoting its vertex. See the figures in this article for examples. The three defining points may also identify angles in geometric figures. For example,
1586-1369: Is worse. If a table or computer implementation of the exsecant function is not available, the exsecant can be accurately computed as exsec θ = tan θ tan 1 2 θ | , {\textstyle \operatorname {exsec} \theta =\tan \theta \,\tan {\tfrac {1}{2}}\theta {\vphantom {\Big |}},} or using versine, exsec θ = vers θ sec θ , {\textstyle \operatorname {exsec} \theta =\operatorname {vers} \theta \,\sec \theta ,} which can itself be computed as vers θ = 2 ( sin 1 2 θ ) ) 2 | = {\textstyle \operatorname {vers} \theta =2{\bigl (}{\sin {\tfrac {1}{2}}\theta }{\bigr )}{\vphantom {)}}^{2}{\vphantom {\Big |}}={}} sin θ tan 1 2 θ | {\displaystyle \sin \theta \,\tan {\tfrac {1}{2}}\theta \,{\vphantom {\Big |}}} ; Haslett used these identities to compute his 1855 exsecant and versine tables. For
1647-578: The Proto-Indo-European root *ank- , meaning "to bend" or "bow". Euclid defines a plane angle as the inclination to each other, in a plane, of two lines that meet each other and do not lie straight with respect to each other. According to the Neoplatonic metaphysician Proclus , an angle must be either a quality, a quantity, or a relationship. The first concept, angle as quality, was used by Eudemus of Rhodes , who regarded an angle as
1708-681: The SI , the radian is treated as being equal to the dimensionless unit 1, thus being normally omitted. The angle expressed by another angular unit may then be obtained by multiplying the angle by a suitable conversion constant of the form k / 2 π , where k is the measure of a complete turn expressed in the chosen unit (for example, k = 360° for degrees or 400 grad for gradians ): θ = k 2 π ⋅ s r . {\displaystyle \theta ={\frac {k}{2\pi }}\cdot {\frac {s}{r}}.} The value of θ thus defined
1769-484: The area of a circle , π r . The other option is to introduce a dimensional constant. According to Quincey this approach is "logically rigorous" compared to SI, but requires "the modification of many familiar mathematical and physical equations". A dimensional constant for angle is "rather strange" and the difficulty of modifying equations to add the dimensional constant is likely to preclude widespread use. In particular, Quincey identifies Torrens' proposal to introduce
1830-446: The cotangent of its complement, and its secant equals the cosecant of its complement.) The prefix " co- " in the names of some trigonometric ratios refers to the word "complementary". If the two supplementary angles are adjacent (i.e., have a common vertex and share just one side), their non-shared sides form a straight line . Such angles are called a linear pair of angles . However, supplementary angles do not have to be on
1891-1276: The Extruding Power of Terrestrial Rotation". Synthese . 134 (1–2, Logic and Mathematical Reasoning): 217–244. doi : 10.1023/A:1022143816001 . JSTOR 20117331 . Searles, William Henry; Ives, Howard Chapin (1915) [1880]. Field Engineering: A Handbook of the Theory and Practice of Railway Surveying, Location and Construction (17th ed.). New York: John Wiley & Sons . Meyer, Carl F. (1969) [1949]. Route Surveying and Design (4th ed.). Scranton, PA: International Textbook Co. "MIT/GNU Scheme – Scheme Arithmetic" ( MIT/GNU Scheme source code). v. 12.1. Massachusetts Institute of Technology . 2023-09-01. exsec function, arith.scm lines 61–63 . Retrieved 2024-04-01 . Review: " Field Manual for Railroad Engineers . By J. C. Nagle" . The Engineer (Review). 84 : 540. 1897-12-03. "MIT/GNU Scheme – Scheme Arithmetic" ( MIT/GNU Scheme source code). v. 12.1. Massachusetts Institute of Technology . 2023-09-01. aexsec function, arith.scm lines 65–71 . Retrieved 2024-04-01 . Complementary angle In Euclidean geometry , an angle
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1952-1091: The US before 1900" . International Journal for the History of Mathematics Education . 6 (2): 55–70. Review: Poor, Henry Varnum , ed. (1856-03-22). " Practical Book of Reference, and Engineer's Field Book . By Charles Haslett" . American Railroad Journal (Review). Second Quarto Series. XII (12): 184. Whole No. 1040, Vol. XXIX. Zucker, Ruth (1964). "4.3.147: Elementary Transcendental Functions - Circular functions" . In Abramowitz, Milton ; Stegun, Irene A. (eds.). Handbook of Mathematical Functions . Washington, D.C.: National Bureau of Standards. p. 78. LCCN 64-60036 . van Haecht, Joannes (1784). "Articulus III: De secantibus circuli: Corollarium III: [109]". Geometria elementaria et practica: quam in usum auditorum (in Latin). Lovanii, e typographia academica. p. 24, foldout. Finocchiaro, Maurice A. (2003). "Physical-Mathematical Reasoning: Galileo on
2013-655: The United States for railroad and road design , and since the early 20th century has sometimes been briefly mentioned in American trigonometry textbooks and general-purpose engineering manuals. For completeness, a few books also defined a coexsecant or excosecant function (symbolized coexsec or excsc ), coexsec θ = {\displaystyle \operatorname {coexsec} \theta ={}} csc θ − 1 , {\displaystyle \csc \theta -1,}
2074-473: The adjacent angles, the vertical angles are equal in measure. According to a historical note, when Thales visited Egypt, he observed that whenever the Egyptians drew two intersecting lines, they would measure the vertical angles to make sure that they were equal. Thales concluded that one could prove that all vertical angles are equal if one accepted some general notions such as: When two adjacent angles form
2135-567: The angle subtended by the circumference of a circle at its centre) is equal to n units, for some whole number n . Two exceptions are the radian (and its decimal submultiples) and the diameter part. In the International System of Quantities , an angle is defined as a dimensionless quantity, and in particular, the radian unit is dimensionless. This convention impacts how angles are treated in dimensional analysis . The following table lists some units used to represent angles. It
2196-457: The angle with vertex A formed by the rays AB and AC (that is, the half-lines from point A through points B and C) is denoted ∠BAC or B A C ^ {\displaystyle {\widehat {\rm {BAC}}}} . Where there is no risk of confusion, the angle may sometimes be referred to by a single vertex alone (in this case, "angle A"). In other ways, an angle denoted as, say, ∠BAC might refer to any of four angles:
2257-1016: The angle): arcexsec y = arcsec ( y + 1 ) = { arctan ( y 2 + 2 y ) if y ≥ 0 , undefined if − 2 < y < 0 , π − arctan ( y 2 + 2 y ) if y ≤ − 2 ; . {\displaystyle \operatorname {arcexsec} y=\operatorname {arcsec}(y+1)={\begin{cases}{\arctan }{\bigl (}\!{\textstyle {\sqrt {y^{2}+2y}}}\,{\bigr )}&{\text{if}}\ \ y\geq 0,\\[6mu]{\text{undefined}}&{\text{if}}\ \ {-2}<y<0,\\[4mu]\pi -{\arctan }{\bigl (}\!{\textstyle {\sqrt {y^{2}+2y}}}\,{\bigr )}&{\text{if}}\ \ y\leq {-2};\\\end{cases}}_{\vphantom {.}}}
2318-893: The arctangent expression is well behaved for small angles. While historical uses of the exsecant did not explicitly involve calculus , its derivative and antiderivative (for x in radians) are: d d x exsec x = tan x sec x , ∫ exsec x d x = ln | sec x + tan x | − x + C , ∫ | {\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {exsec} x&=\tan x\,\sec x,\\[10mu]\int \operatorname {exsec} x\,\mathrm {d} x&=\ln {\bigl |}\sec x+\tan x{\bigr |}-x+C,{\vphantom {\int _{|}}}\end{aligned}}} where ln
2379-497: The area of a circular sector θ = 2 A / r gives 1 radian as 1 m /m = 1. The key fact is that the radian is a dimensionless unit equal to 1 . In SI 2019, the SI radian is defined accordingly as 1 rad = 1 . It is a long-established practice in mathematics and across all areas of science to make use of rad = 1 . Giacomo Prando writes "the current state of affairs leads inevitably to ghostly appearances and disappearances of
2440-594: The central angle between the segment's inner endpoint and the point of tangency for a line through the outer endpoint and tangent to the circle. The word secant comes from Latin for "to cut", and a general secant line "cuts" a circle, intersecting it twice; this concept dates to antiquity and can be found in Book 3 of Euclid's Elements , as used e.g. in the intersecting secants theorem . 18th century sources in Latin called any non- tangential line segment external to
2501-407: The clockwise angle from B to C about A, the anticlockwise angle from B to C about A, the clockwise angle from C to B about A, or the anticlockwise angle from C to B about A, where the direction in which the angle is measured determines its sign (see § Signed angles ). However, in many geometrical situations, it is evident from the context that the positive angle less than or equal to 180 degrees
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2562-407: The difference between two approximately equal quantities results in catastrophic cancellation : because most of the digits of each quantity are the same, they cancel in the subtraction, yielding a lower-precision result. For example, the secant of 1° is sec 1° ≈ 1.000 152 , with the leading several digits wasted on zeros, while the common logarithm of the exsecant of 1°
2623-429: The ends of the arc, which equals the radius times the trigonometric exsecant of half the central angle subtended by the arc, R exsec 1 2 Δ . {\displaystyle R\operatorname {exsec} {\tfrac {1}{2}}\Delta .} By comparison, the versed sine of a curved track section is the furthest distance from the long chord (the line segment between endpoints) to
2684-478: The expressions 1 − cos θ {\displaystyle 1-\cos \theta } (versine) and sec θ − 1 {\displaystyle \sec \theta -1} (exsecant) is problematic for small angles where sec θ ≈ cos θ ≈ 1. {\displaystyle \sec \theta \approx \cos \theta \approx 1.} Computing
2745-421: The exsecant of the complementary angle , though it was not used in practice. While the exsecant has occasionally found other applications, today it is obscure and mainly of historical interest. As a line segment , an external secant of a circle has one endpoint on the circumference, and then extends radially outward. The length of this segment is the radius of the circle times the trigonometric exsecant of
2806-450: The final position is the same, a physical rotation (movement) of −45° is not the same as a rotation of 315° (for example, the rotation of a person holding a broom resting on a dusty floor would leave visually different traces of swept regions on the floor). In three-dimensional geometry, "clockwise" and "anticlockwise" have no absolute meaning, so the direction of positive and negative angles must be defined in terms of an orientation , which
2867-554: The late-19th and 20th century, railroads began using arcs of an Euler spiral as a track transition curve between straight or circular sections of differing curvature. These spiral curves can be approximately calculated using exsecants and versines. Solving the same types of problems is required when surveying circular sections of canals and roads, and the exsecant was still used in mid-20th century books about road surveying. The exsecant has sometimes been used for other applications, such as beam theory and depth sounding with
2928-485: The logarithm of the exsecant and versine saved significant effort and produced more accurate results compared to calculating the same quantity from values found in previously available trigonometric tables. The same idea was adopted by other authors, such as Searles (1880). By 1913 Haslett's approach was so widely adopted in the American railroad industry that, in that context, "tables of external secants and versed sines [were] more common than [were] tables of secants". In
2989-565: The measure of Angle B . Using the measure of either angle C or angle D , we find the measure of angle B to be 180° − (180° − x ) = 180° − 180° + x = x . Therefore, both angle A and angle B have measures equal to x and are equal in measure. A transversal is a line that intersects a pair of (often parallel) lines and is associated with exterior angles , interior angles , alternate exterior angles , alternate interior angles , corresponding angles , and consecutive interior angles . The angle addition postulate states that if B
3050-459: The name secant . In the 19th century, most railroad tracks were constructed out of arcs of circles , called simple curves . Surveyors and civil engineers working for the railroad needed to make many repetitive trigonometrical calculations to measure and plan circular sections of track. In surveying, and more generally in practical geometry, tables of both "natural" trigonometric functions and their common logarithms were used, depending on
3111-467: The negative y -axis. When Cartesian coordinates are represented by standard position , defined by the x -axis rightward and the y -axis upward, positive rotations are anticlockwise , and negative cycles are clockwise . In many contexts, an angle of − θ is effectively equivalent to an angle of "one full turn minus θ ". For example, an orientation represented as −45° is effectively equal to an orientation defined as 360° − 45° or 315°. Although
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#17327983258453172-437: The quantities of torque (N⋅m) and angular momentum (kg⋅m /s). At least a dozen scientists between 1936 and 2022 have made proposals to treat the radian as a base unit of measurement for a base quantity (and dimension) of "plane angle". Quincey's review of proposals outlines two classes of proposal. The first option changes the unit of a radius to meters per radian, but this is incompatible with dimensional analysis for
3233-404: The radian in the dimensional analysis of physical equations". For example, an object hanging by a string from a pulley will rise or drop by y = rθ centimetres, where r is the magnitude of the radius of the pulley in centimetres and θ is the magnitude of the angle through which the pulley turns in radians. When multiplying r by θ , the unit radian does not appear in the product, nor does
3294-582: The same line and can be separated in space. For example, adjacent angles of a parallelogram are supplementary, and opposite angles of a cyclic quadrilateral (one whose vertices all fall on a single circle) are supplementary. If a point P is exterior to a circle with center O, and if the tangent lines from P touch the circle at points T and Q, then ∠TPQ and ∠TOQ are supplementary. The sines of supplementary angles are equal. Their cosines and tangents (unless undefined) are equal in magnitude but have opposite signs. In Euclidean geometry, any sum of two angles in
3355-550: The sides. In the case of a rotation , the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation. The word angle comes from the Latin word angulus , meaning "corner". Cognate words include the Greek ἀγκύλος ([ankylοs] Error: {{Lang}}: Non-latn text/Latn script subtag mismatch ( help ) ) meaning "crooked, curved" and the English word " ankle ". Both are connected with
3416-404: The specific calculation. Using logarithms converts expensive multiplication of multi-digit numbers to cheaper addition, and logarithmic versions of trigonometric tables further saved labor by reducing the number of necessary table lookups. The external secant or external distance of a curved track section is the shortest distance between the track and the intersection of the tangent lines from
3477-980: The summation of angles: The adjective complementary is from the Latin complementum , associated with the verb complere , "to fill up". An acute angle is "filled up" by its complement to form a right angle. The difference between an angle and a right angle is termed the complement of the angle. If angles A and B are complementary, the following relationships hold: sin 2 A + sin 2 B = 1 cos 2 A + cos 2 B = 1 tan A = cot B sec A = csc B {\displaystyle {\begin{aligned}&\sin ^{2}A+\sin ^{2}B=1&&\cos ^{2}A+\cos ^{2}B=1\\[3pt]&\tan A=\cot B&&\sec A=\csc B\end{aligned}}} (The tangent of an angle equals
3538-422: The table below: When two straight lines intersect at a point, four angles are formed. Pairwise, these angles are named according to their location relative to each other. The equality of vertically opposite angles is called the vertical angle theorem . Eudemus of Rhodes attributed the proof to Thales of Miletus . The proposition showed that since both of a pair of vertical angles are supplementary to both of
3599-474: The track – cf. Sagitta – which equals the radius times the trigonometric versine of half the central angle, R vers 1 2 Δ . {\displaystyle R\operatorname {vers} {\tfrac {1}{2}}\Delta .} These are both natural quantities to measure or calculate when surveying circular arcs, which must subsequently be multiplied or divided by other quantities. Charles Haslett (1855) found that directly looking up
3660-489: The unit centimetre—because both factors are magnitudes (numbers). Similarly in the formula for the angular velocity of a rolling wheel, ω = v / r , radians appear in the units of ω but not on the right hand side. Anthony French calls this phenomenon "a perennial problem in the teaching of mechanics". Oberhofer says that the typical advice of ignoring radians during dimensional analysis and adding or removing radians in units according to convention and contextual knowledge
3721-437: Was introduced in 1855 by American civil engineer Charles Haslett , who used it in conjunction with the existing versine function, vers θ = 1 − cos θ , {\displaystyle \operatorname {vers} \theta =1-\cos \theta ,} for designing and measuring circular sections of railroad track. It was adopted by surveyors and civil engineers in
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