CHAPTER 21NAVIGATIONALMATHEMATICSGEOMETRY2100.Definitionapoint. It is measured by the arc of a circle intercepted between the two linesforming the angle,thecenter ofthe circlebeing at the point of intersection.In Figure 2101, the angleGeometry deals with the properties, relations,andformed bylines ABand BC,may be designated“angleB,"measurementof lines,surfaces,solids,and angles.Plane"angle ABC"or“angle CBA";or by Greek letter as“anglegeometry deals with plane figures, and solid geometryα."The three letterdesignation is preferred if there is moredeals with three-dimensional figures.than one angle at the point. When three letters are used, theApoint,consideredmathematically,is a placehavingmiddle one should always be that at the vertex of the angleposition but no extent. It has no length, breadth, or thick-An acute angle is one less than a right angle (900)ness.Apoint in motion producesa line,which has lengthA right angle is one whose sides are perpendicular (900)but neither breadth nor thickness. A straight or right lineAn obtuse angle is one greater than a right angle (900)istheshortestdistancebetweentwopointsinspace.Alinebutless than1800in motion in anydirection except along itself producesaA straight angle is one whose sides form a continuoussurface.whichhaslengthandbreadthbutnotthickness.Astraight line (180°)plane surface or plane is a surface without curvature. AA reflex angle is one greater than a straight anglestraight line connecting any twoof its points lies wholly(180°)but less than a circle (360°).Any two lines meetingwithintheplane.Aplanesurfaceinmotion inanydirectionatapointform two angles, one less thana straight angleofexcept within its plane produces a solid, which has length,180(unless exactly a straightangle)and the othergreaterbreadth.andthickness.Parallellinesorsurfacesarethosethan a straightangle.which are everywhere equidistant. Perpendicular lines orAn oblique angle is any angle not a multiple of 900surfaces are those which meet at right or 90°angles.A per-Two angles whosesum isaright angle(90°)are com-pendicular may be called a normal, particularly when it isplementary angles,and either is the complement of theperpendicular to the tangent to a curved line or surface atother.thepoint oftangency.Allpoints equidistantfromtheendsTwo angles whose sum is a straight angle (180°) areof a straight line are on the perpendicular bisector of thatsupplementary angles, and either is the supplement oftheline.Theshortestdistancefrom apointtoa line is thelengthother.oftheperpendicularbetweenthemTwoangleswhose sumisa circle(360°)are exple-mentary angles, and either is the explement of the other2101.AnglesThetwo angles formed when any two lines terminate at acommonpointareexplementaryAn angleis formedbytwo straight lines whichmeetatIf the sides of one angle are perpendicular to those ofanother,thetwoanglesare eitherequal or supplementary.Also, if the sides ofone angle are parallel to those ofanoth-er, the two angles are either equal or supplementary.Whentwo straight lines intersect,formingfourangles.the two opposite angles, called vertical angles, are equalAngles which have the same vertex and lie on oppositesides of a common side are adjacent angles. Adjacent an-gles formed by intersecting lines are supplementary, sinceeachpair of adjacent angles forms a straight angle.Atransversalis a linethat intersects two ormoreotherlines.If two ormoreparallel lines are cut by a transversal.groups of adjacentandverticalangles areformed,CA dihedral angle is the anglebetween two intersectingFigure 2101. An angle.planes.327
327 CHAPTER 21 NAVIGATIONAL MATHEMATICS GEOMETRY 2100. Definition Geometry deals with the properties, relations, and measurement of lines, surfaces, solids, and angles. Plane geometry deals with plane figures, and solid geometry deals with three–dimensional figures. A point, considered mathematically, is a place having position but no extent. It has no length, breadth, or thickness. A point in motion produces a line, which has length, but neither breadth nor thickness. A straight or right line is the shortest distance between two points in space. A line in motion in any direction except along itself produces a surface, which has length and breadth, but not thickness. A plane surface or plane is a surface without curvature. A straight line connecting any two of its points lies wholly within the plane. A plane surface in motion in any direction except within its plane produces a solid, which has length, breadth, and thickness. Parallel lines or surfaces are those which are everywhere equidistant. Perpendicular lines or surfaces are those which meet at right or 90° angles. A perpendicular may be called a normal, particularly when it is perpendicular to the tangent to a curved line or surface at the point of tangency. All points equidistant from the ends of a straight line are on the perpendicular bisector of that line. The shortest distance from a point to a line is the length of the perpendicular between them. 2101. Angles An angle is formed by two straight lines which meet at a point. It is measured by the arc of a circle intercepted between the two lines forming the angle, the center of the circle being at the point of intersection. In Figure 2101, the angle formed by lines AB and BC, may be designated “angle B,” “angle ABC,” or “angle CBA”; or by Greek letter as “angle α.” The three letter designation is preferred if there is more than one angle at the point. When three letters are used, the middle one should always be that at the vertex of the angle. An acute angle is one less than a right angle (90°). A right angle is one whose sides are perpendicular (90°). An obtuse angle is one greater than a right angle (90°) but less than 180°. A straight angle is one whose sides form a continuous straight line (180°). A reflex angle is one greater than a straight angle (180°) but less than a circle (360°). Any two lines meeting at a point form two angles, one less than a straight angle of 180° (unless exactly a straight angle) and the other greater than a straight angle. An oblique angle is any angle not a multiple of 90°. Two angles whose sum is a right angle (90°) are complementary angles, and either is the complement of the other. Two angles whose sum is a straight angle (180°) are supplementary angles, and either is the supplement of the other. Two angles whose sum is a circle (360°) are explementary angles, and either is the explement of the other. The two angles formed when any two lines terminate at a common point are explementary. If the sides of one angle are perpendicular to those of another, the two angles are either equal or supplementary. Also, if the sides of one angle are parallel to those of another, the two angles are either equal or supplementary. When two straight lines intersect, forming four angles, the two opposite angles, called vertical angles, are equal. Angles which have the same vertex and lie on opposite sides of a common side are adjacent angles. Adjacent angles formed by intersecting lines are supplementary, since each pair of adjacent angles forms a straight angle. A transversal is a line that intersects two or more other lines. If two or more parallel lines are cut by a transversal, groups of adjacent and vertical angles are formed, A dihedral angle is the angle between two intersecting Figure 2101. An angle. planes
328NAVIGATIONALMATHEMATICS2102.Trianglescenter of the opposite side.The three medians of a trianglemeet at a point called the centroid ofthe triangle. This pointAplane triangle is a closed figure formed by threedivides each median into two parts, that part between thecentroid and thevertex beingtwiceas long as the otherpart.straight lines, called sides, which meet at threepoints calledvertices.Theverticesarelabeled withcapital letters andtheLines bisecting the three angles of a triangle meet at asides with lowercase letters,as shown in Figure 2102apoint which is equidistantfrom the three sides, which is theAn equilateral triangle is one with its three sidescenter of the inscribed circle, as shown in Figure 2102b.equal in length.It must also be equiangular, with its threeThis point is of particular interest tonavigators because it isangles equal.thepointtheoreticallytakenasthefixwhenthreelinesofAn isosceles triangle is one with two equal sides.positionof equal weightand having onlyrandom errorsdonotmeetata common point.In practical navigation,thecalled legs.Theangles oppositethelegs are equal.A linewhich bisects (divides into two equal parts) the unequal an-point is found visually,notby construction,and otherfac-tors often influencethe chosen fixposition.gle ofan isoscelestriangleistheperpendicularbisectorofthe opposite side, and divides the triangle into two equalTheperpendicular bisectors of the three sides of a tri-right triangles.angle meet at a point which is equidistant from the threeA scalene triangle is one with no two sides equal. Invertices.whichisthe center of the circumscribed circlesuchatriangle,notwo angles areequalthe circle through the three vertices and the smallest circlewhich can be drawn enclosing thetriangle.The center of aAn acute triangle is one with three acute angles.circumscribed circle is within an acute triangle, on the hyA right triangle is one having a right angle.The sidepotenuse ofaright triangle, and outside an obtuse triangleopposite the right angleis called thehypotenuse.The othertwo sides may becalled legs.Aplane trianglecan have onlyA line connecting themid-points of two sides ofatri-angle is always parallel to the third side and half as longoneright angle.Also,a lineparalleltoonesideofatriangleand intersectingAn obtuse triangle is one with an obtuse angle. Athe other two sides divides these sides proportionally.Thisplane triangle can have only one obtuse angle.principle can be used to divide a line into any number ofAn oblique triangle is onewhich doesnot contain aequalorproportionalpartsright angleThe sum of the angles of a plane triangle is alwaysThe altitude of a triangle is a line or the distance from180oTherefore, the sum of theacute angles ofa right tri-anyvertexperpendiculartotheopposite side.angle is 90°,and theanglesare complementary.Ifone sideA median of a triangle is a linefrom any vertex to theof atriangle is extended, the exterior angle thus formed issupplementary to the adjacent interior angle and is therefore equal to the sum of thetwo non adjacent angles.If twoangles of one triangleareequal to twoangles ofanothertri-Bangle, the third angles are also equal, and the triangles aresimilar. If the area of one triangle is equal to the area of an-other, the triangles are equal.Triangles having equal basesand altitudes also have equal areas.Two figures are con-gruent ifonecanbeplacedovertheothertomakeanexactfit.Congruent figures are bothsimilar and equal.Ifanysideofone triangle is equal toany sideofa similartriangle,theDtriangles are congruent. For example, if two right trianglesbhave equal sides, they are congruent, if two right triangleshavetwo corresponding sides equal, they are congruent.Figure 2102a.Atriangle.Triangles are congruent only if the sides and angles areequal.The sum of two sides of aplanetriangle is alwaysgreater than the third side;their difference is always lessthanthethird sideThe area of a triangle is equal to 1/2 of the area of thepolygon formed from its baseand height.This can be statedalgebraicallyas:bhAreaofplanetriangleA=2The square ofthe hypotenuse ofaright triangle is equalto the sum of the squares of the other two sides, or a2 + b2Figure2102b.A circle inscribed in a triangle
328 NAVIGATIONAL MATHEMATICS 2102. Triangles A plane triangle is a closed figure formed by three straight lines, called sides, which meet at three points called vertices. The vertices are labeled with capital letters and the sides with lowercase letters, as shown in Figure 2102a. An equilateral triangle is one with its three sides equal in length. It must also be equiangular, with its three angles equal. An isosceles triangle is one with two equal sides, called legs. The angles opposite the legs are equal. A line which bisects (divides into two equal parts) the unequal angle of an isosceles triangle is the perpendicular bisector of the opposite side, and divides the triangle into two equal right triangles. A scalene triangle is one with no two sides equal. In such a triangle, no two angles are equal. An acute triangle is one with three acute angles. A right triangle is one having a right angle. The side opposite the right angle is called the hypotenuse. The other two sides may be called legs. A plane triangle can have only one right angle. An obtuse triangle is one with an obtuse angle. A plane triangle can have only one obtuse angle. An oblique triangle is one which does not contain a right angle. The altitude of a triangle is a line or the distance from any vertex perpendicular to the opposite side. A median of a triangle is a line from any vertex to the center of the opposite side. The three medians of a triangle meet at a point called the centroid of the triangle. This point divides each median into two parts, that part between the centroid and the vertex being twice as long as the other part. Lines bisecting the three angles of a triangle meet at a point which is equidistant from the three sides, which is the center of the inscribed circle, as shown in Figure 2102b. This point is of particular interest to navigators because it is the point theoretically taken as the fix when three lines of position of equal weight and having only random errors do not meet at a common point. In practical navigation, the point is found visually, not by construction, and other factors often influence the chosen fix position. The perpendicular bisectors of the three sides of a triangle meet at a point which is equidistant from the three vertices, which is the center of the circumscribed circle, the circle through the three vertices and the smallest circle which can be drawn enclosing the triangle. The center of a circumscribed circle is within an acute triangle, on the hypotenuse of a right triangle, and outside an obtuse triangle. A line connecting the mid–points of two sides of a triangle is always parallel to the third side and half as long. Also, a line parallel to one side of a triangle and intersecting the other two sides divides these sides proportionally. This principle can be used to divide a line into any number of equal or proportional parts. The sum of the angles of a plane triangle is always 180°. Therefore, the sum of the acute angles of a right triangle is 90°, and the angles are complementary. If one side of a triangle is extended, the exterior angle thus formed is supplementary to the adjacent interior angle and is therefore equal to the sum of the two non adjacent angles. If two angles of one triangle are equal to two angles of another triangle, the third angles are also equal, and the triangles are similar. If the area of one triangle is equal to the area of another, the triangles are equal. Triangles having equal bases and altitudes also have equal areas. Two figures are congruent if one can be placed over the other to make an exact fit. Congruent figures are both similar and equal. If any side of one triangle is equal to any side of a similar triangle, the triangles are congruent. For example, if two right triangles have equal sides, they are congruent; if two right triangles have two corresponding sides equal, they are congruent. Triangles are congruent only if the sides and angles are equal. The sum of two sides of a plane triangle is always greater than the third side; their difference is always less than the third side. The area of a triangle is equal to 1/2 of the area of the polygon formed from its base and height. This can be stated algebraically as: The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides, or a2 + b2 Figure 2102a. A triangle. Figure 2102b. A circle inscribed in a triangle. Area of plane triangle A = bh 2 -
329NAVIGATIONALMATHEMATICSthe circle,whichhas onlyone point in common with the cir-=c2.Therefore the length of the hypotenuse of plane rightcumference. A tangent is perpendicular to the radius at thetrianglecanbefound bytheformula:point of tangency.Two tangents from a common point toopposite sidesofa circleareequal inlength,anda linefromc = a2+b2the point to the center of the circle bisects the angleformedby the two tangents.An angle formed outside a circleby the2103.Circlesintersection oftwo tangents, a tangent and a secant, or twosecants has half as manydegrees as the differencebetweenAcircle isaplane,closed curve,all pointsofwhich arethetwointerceptedarcs.Anangleformedbyatangentandequidistant from a point within, called the center.a chord,with theapexatthepoint oftangency,hashalf asThe distance around a circle is called the circumfermany degrees as the arc it intercepts. A common tangentis one tangent to more than one circle.Two circles are tan-ence. Technically the length of this line is the perimeteralthough theterm“circumference"is often used.An arc isgent to each other if they touch at one point only.If ofpartofa circumference. Amajor arc is more than a semicir-differentsizes,thesmallercirclemaybeeitherinsideoroutsidethe largerone.cle(180),aminorareislessthanasemicircle(180°).Asemi-circle is half a circle (180°), a quadrant is a quarterParallel lines intersecting a circle intercept equal arcs.ofacircle(90°),aquintantisafifthofacircle(72),a sex-tant is a sixth of a circle (60°),an octant is an eighth of aIf A= area, r = radius, d = diameter, C = circumfer-circle (45°). Some of these names have been applied to in-ence;s=linear lengthofan arca=angular lengthofanarc,strumentsused bynavigatorsformeasuringaltitudes ofor the angle it subtends at the center of a circle, in degrees;celestial bodies because of the part of a circle used for theb = angular length of an arc, or the angle it subtends at thelength ofthearc of the instrument.center of a circle,in radians:Concentric circles have a common center.A radius(plural radi)or semidiameter is a straight line connectingArea ofcircle A = nr?nd?thecenterofa circle withany point on its circumference.4Adiameterofacircle is a straight line passing throughitscenterand terminating at oppositesides ofthecircumfer-CircumferenceofacircleC=2元r=元d=2元radence.Itdivides acircle into twoequal parts.Theratio of the元a-2b_rslengthofthecircumferenceof anycircletothelengthofitsAreaofsector360diameter is 3.14159+,or π(the Greek letter pi),a relation-?(b- sina)shipthat hasmanyuseful applications.Areaofsegment=A sector is that part of a circle bounded by two radi2and an arc.Theangleformed bytwo radi is called a cen-tral angle. Any pair of radii divides a circle into sectors,2104.Spheresonelessthanasemicircle(180)andtheothergreaterthana semicircle (unless thetwo radiiform a diameter)A sphere is a solid bounded by a surface every point ofA chord is a straight line connecting any two points onwhich is equidistantfrom a point within calledthe center.Itthe circumference ofa circle.Chords equidistant from themay alsobe formed byrotating a circle about any diameter.center of a circle are equal in length.A radius or semidiameterofa sphere is a straight lineA segment is the part of a circle bounded by a chordconnecting its centerwithanypoint on its surface.Adiam-and the intercepted arc.A chord divides a circle into twoeter of a sphere is a straight line through its center andsegments,oneless than a semicircle(180°),and theotherterminated at both ends bythe surfaceof the spheregreater than a semicircle (unless the chord is a diameter).AThe intersection of a plane and the surface of a spherediameter perpendicularto a chord bisects it, its arc, and itsis a circle,a great circle if the plane passes through the cen-segments.Either pair ofvertical angles formed by intersect-ter of the sphere,and a small circle ifit doesnot.The shortering chords has a combined numberof degrees equal to thearc of thegreat circlebetweentwopoints on the surfaceofasum of the numberofdegrees inthetwo arcs intercepted bysphere is the shortest distance, on the surface of the sphere,the two angles.between the points.Everygreatcircle of a sphere bisects ev-An inscribed angle is one whose vertex is on the cir-ery other great circle of that sphere.Thepoles of a circle oncumference of a circleand whose sides are chords.It hasa sphere are the extremities of the sphere's diameter whichhalfasmany degrees as the arc it intercepts.Hence,an angleis perpendicular to theplane ofthe circle.All points on theinscribed in a semicircle isa right angle if its sidesterminatecircumference of the circle are equidistant from either of itsat theends of thediameterformingthesemicirclepoles.In the ease of a great circle, both poles are 90°fromA secant of a circle is a line intersecting the circle, orany point on the circumference ofthecircle.Any greatcircleachord extendedbeyondthecircumferencemaybe considered a primary,particularlywhen it serves asA tangent to a circle is a straight line, in the plane ofthe origin ofmeasurement ofa coordinate.Thegreat circles
NAVIGATIONAL MATHEMATICS 329 = c2. Therefore the length of the hypotenuse of plane right triangle can be found by the formula : 2103. Circles A circle is a plane, closed curve, all points of which are equidistant from a point within, called the center. The distance around a circle is called the circumference. Technically the length of this line is the perimeter, although the term “circumference” is often used. An arc is part of a circumference. A major arc is more than a semicircle (180°), a minor are is less than a semicircle (180°). A semi–circle is half a circle (180°), a quadrant is a quarter of a circle (90°), a quintant is a fifth of a circle (72°), a sextant is a sixth of a circle (60°), an octant is an eighth of a circle (45°). Some of these names have been applied to instruments used by navigators for measuring altitudes of celestial bodies because of the part of a circle used for the length of the arc of the instrument. Concentric circles have a common center. A radius (plural radii) or semidiameter is a straight line connecting the center of a circle with any point on its circumference. A diameter of a circle is a straight line passing through its center and terminating at opposite sides of the circumference. It divides a circle into two equal parts. The ratio of the length of the circumference of any circle to the length of its diameter is 3.14159+, or π (the Greek letter pi), a relationship that has many useful applications. A sector is that part of a circle bounded by two radii and an arc. The angle formed by two radii is called a central angle. Any pair of radii divides a circle into sectors, one less than a semicircle (180°) and the other greater than a semicircle (unless the two radii form a diameter). A chord is a straight line connecting any two points on the circumference of a circle. Chords equidistant from the center of a circle are equal in length. A segment is the part of a circle bounded by a chord and the intercepted arc. A chord divides a circle into two segments, one less than a semicircle (180°), and the other greater than a semicircle (unless the chord is a diameter). A diameter perpendicular to a chord bisects it, its arc, and its segments. Either pair of vertical angles formed by intersecting chords has a combined number of degrees equal to the sum of the number of degrees in the two arcs intercepted by the two angles. An inscribed angle is one whose vertex is on the circumference of a circle and whose sides are chords. It has half as many degrees as the arc it intercepts. Hence, an angle inscribed in a semicircle is a right angle if its sides terminate at the ends of the diameter forming the semicircle. A secant of a circle is a line intersecting the circle, or a chord extended beyond the circumference. A tangent to a circle is a straight line, in the plane of the circle, which has only one point in common with the circumference. A tangent is perpendicular to the radius at the point of tangency. Two tangents from a common point to opposite sides of a circle are equal in length, and a line from the point to the center of the circle bisects the angle formed by the two tangents. An angle formed outside a circle by the intersection of two tangents, a tangent and a secant, or two secants has half as many degrees as the difference between the two intercepted arcs. An angle formed by a tangent and a chord, with the apex at the point of tangency, has half as many degrees as the arc it intercepts. A common tangent is one tangent to more than one circle. Two circles are tangent to each other if they touch at one point only. If of different sizes, the smaller circle may be either inside or outside the larger one. Parallel lines intersecting a circle intercept equal arcs. If A = area; r = radius; d = diameter; C = circumference; s = linear length of an arc; a = angular length of an arc, or the angle it subtends at the center of a circle, in degrees; b = angular length of an arc, or the angle it subtends at the center of a circle, in radians: 2104. Spheres A sphere is a solid bounded by a surface every point of which is equidistant from a point within called the center. It may also be formed by rotating a circle about any diameter. A radius or semidiameter of a sphere is a straight line connecting its center with any point on its surface. A diameter of a sphere is a straight line through its center and terminated at both ends by the surface of the sphere. The intersection of a plane and the surface of a sphere is a circle, a great circle if the plane passes through the center of the sphere, and a small circle if it does not. The shorter arc of the great circle between two points on the surface of a sphere is the shortest distance, on the surface of the sphere, between the points. Every great circle of a sphere bisects every other great circle of that sphere. The poles of a circle on a sphere are the extremities of the sphere’s diameter which is perpendicular to the plane of the circle. All points on the circumference of the circle are equidistant from either of its poles. In the ease of a great circle, both poles are 90° from any point on the circumference of the circle. Any great circle may be considered a primary, particularly when it serves as the origin of measurement of a coordinate. The great circles c a 2 b 2 = + Area of circle A πr 2 πd2 4 = = - Circumference of a circle C 2 = == πr πd 2π rad Area of sector πr 2a 360 - r 2b 2 - rs 2 = == - Area of segment r 2(b a – sin ) 2 = -
330NAVIGATIONAL MATHEMATICSthrough its poles are called secondary.Secondaries are per-(called thepole).Aline extending in the direction indicatedpendicular to theirprimaryis called a radius vector.Direction and distance from aA spherical triangle is the figure formed on the sur-fixed point constitutepolarcoordinates, sometimes calledface of a sphere by the intersection of three great circles.the rho-theta (the Greek p, to indicate distance, and theThe lengths of the sides ofa spherical triangle are measuredGreek ,toindicatedirection)system.An example of itsin degrees,minutes, and seconds, as the angular lengths ofuse isthe radar scope.the arcsformingthem.The sumof thethreesides is alwaysSpherical coordinates are used to define a position onless than360o.The sumof thethreeangles is always morethe surface of a sphere or spheroid by indicating angularthan180°andlessthan540°distancefrom a primarygreat circleandareferencesecond-Alune is thepart of the surfaceofa sphereboundedbyary great circle. Examples used in navigation are latitudehalves of twogreat circles.and longitude, altitude and azimuth, and declination andhourangle.2105.Co0rdinatesCoordinates aremagnitudes used to defineapositionManydifferenttypes of coordinates areused.Importantnavigational ones aredescribedbelow.if a position is known to be on a given line,only onemagnitude(coordinate)is neededtoidentifythepositionifYan origin is stated or understood.If a position is known to be on a given surface, twomagnitudes (coordinates)are neededto definethe position.If nothing isknown regarding aposition other than thatitexists inspace,three magnitudes (coordinates)are neededto define its position.AEach coordinate requires an origin,either stated or im-plied.Ifaposition isknownto beonagivenplane, itmightbedefinedbymeansof its distancefromeachoftwo inter-secting lines, called axes. These are called rectangularcoordinates.InFigure2105, OY is called the ordinate,Jand OX is called the abscissa. Point O is the origin, andlines OX and OY the axes (called theX and Y axes, respec-tively).Point A is at position x,y.If the axes arenotperpendicular but the lines x and y aredrawn parallel to theaxes, oblique coordinates result. Either type are calledXCartesian coordinates. A three-dimensional system ofCartesiancoordinates,withXY,andZaxes,is calledspacecoordinates.Anothersystemof planecoordinates incommon usageFigure 2105. Rectangular coordinates.consists of the direction and distance from the originTRIGONOMETRY2106.Definitions2107.AngularMeasureTrigonometry deals with the relations among the an-A circle may be divided into 360 degrees (°), which isgles and sides of triangles.Plane trigonometry deals withthe angular length of its circumference.Each degreemayplane triangles, those on a plane surface. Spherical trigo-be divided into 60minutes ), and eachminute into 60 sec-nometry deals with spherical triangles, which are drawn ononds (").The angular length of an arc is usually expressedthe surfaceofa sphere.In navigation,the commonmethodsin these units.By this system a right angle or quadrant hasof celestial sight reduction use spherical triangles on the sur-90°and a straightangleor semicircle1800In marine nav-faceof the earth.Formost navigational purposes,the earthigation, altitudes, latitudes, and longitudes are usuallyis assumed tobe a sphere,thoughit is somewhatflattened.expressed in degrees, minutes, and tenths (27°14.4).Azi-
330 NAVIGATIONAL MATHEMATICS through its poles are called secondary. Secondaries are perpendicular to their primary. A spherical triangle is the figure formed on the surface of a sphere by the intersection of three great circles. The lengths of the sides of a spherical triangle are measured in degrees, minutes, and seconds, as the angular lengths of the arcs forming them. The sum of the three sides is always less than 360°. The sum of the three angles is always more than 180° and less than 540°. A lune is the part of the surface of a sphere bounded by halves of two great circles. 2105. Coordinates Coordinates are magnitudes used to define a position. Many different types of coordinates are used. Important navigational ones are described below. If a position is known to be on a given line, only one magnitude (coordinate) is needed to identify the position if an origin is stated or understood. If a position is known to be on a given surface, two magnitudes (coordinates) are needed to define the position. If nothing is known regarding a position other than that it exists in space, three magnitudes (coordinates) are needed to define its position. Each coordinate requires an origin, either stated or implied. If a position is known to be on a given plane, it might be defined by means of its distance from each of two intersecting lines, called axes. These are called rectangular coordinates. In Figure 2105, OY is called the ordinate, and OX is called the abscissa. Point O is the origin, and lines OX and OY the axes (called the X and Y axes, respectively). Point A is at position x,y. If the axes are not perpendicular but the lines x and y are drawn parallel to the axes, oblique coordinates result. Either type are called Cartesian coordinates. A three–dimensional system of Cartesian coordinates, with X Y, and Z axes, is called space coordinates. Another system of plane coordinates in common usage consists of the direction and distance from the origin (called the pole). A line extending in the direction indicated is called a radius vector. Direction and distance from a fixed point constitute polar coordinates, sometimes called the rho–theta (the Greek ρ, to indicate distance, and the Greek θ, to indicate direction) system. An example of its use is the radar scope. Spherical coordinates are used to define a position on the surface of a sphere or spheroid by indicating angular distance from a primary great circle and a reference secondary great circle. Examples used in navigation are latitude and longitude, altitude and azimuth, and declination and hour angle. TRIGONOMETRY 2106. Definitions Trigonometry deals with the relations among the angles and sides of triangles. Plane trigonometry deals with plane triangles, those on a plane surface. Spherical trigonometry deals with spherical triangles, which are drawn on the surface of a sphere. In navigation, the common methods of celestial sight reduction use spherical triangles on the surface of the earth. For most navigational purposes, the earth is assumed to be a sphere, though it is somewhat flattened. 2107. Angular Measure A circle may be divided into 360 degrees (°), which is the angular length of its circumference. Each degree may be divided into 60 minutes ('), and each minute into 60 seconds ("). The angular length of an arc is usually expressed in these units. By this system a right angle or quadrant has 90° and a straight angle or semicircle 180°. In marine navigation, altitudes, latitudes, and longitudes are usually expressed in degrees, minutes, and tenths (27°14.4'). AziFigure 2105. Rectangular coordinates
331NAVIGATIONALMATHEMATICSmuths are usually expressed in degrees and tenths (164.70)ciprocals of the first three; thereforeThe system of degrees, minutes, and seconds indicatedR-Aabove is the sexagesimal system.In the centesimal system-Pused chiefly in France, the circle is divided into 400 centes-11imal degrees (sometimes called grades)each of which is-!divided into100centesimal minutes of 100centesimal sec---onds each.1-Aradian is the angle subtended at the center ofa circle1-by an arc having a linear length equal to the radius of the11circle.Acircle(360°)=2元 radians,a semicircle(180°)=元18radians, a right angle (90°)=/2 radians. The length of theA0-Barc ofa circle is equal to the radius multiplied by the angleFsubtended in radians.2108.TrigonometricFunctionsFigure 2108a. Similar right triangles.Trigonometric functions are the various proportionsor ratios of the sides of a plane right triangle, defined in re-11csce =sing=lation to one of the acute angles.In Figure2108a, letbecscesineany acute angle. From any point R on line OA, draw a line11coso=sece=perpendicular to OB at F.From any other point Ron OA,sececosedrawa lineperpendicular to OBat F:Then triangles OFR11and OF'R'are similar right trianglesbecauseall their corre-tane=cote=cotetanesponding angles are equal. Since in any pair of similartriangles the ratio ofany two sides ofone triangle is equal toIn Figure 2108b, A, B,and C are the angles ofaplanetheratioofthe correspondingtwosides oftheothertriangle,right triangle, with the right angle at C. The sides are la-OF_OF"RFRFR'Fbeled a, b, c, opposite angles A, B, and C respectively. The(RF,OF)=andOF=OROR1ORORsix principaltrigonometricfunctions of angle B are:b1oNomatterwhere thepoint Rislocated onOA, theratio1sin B=COS A=cos(90°B)between the lengths of any two sides in the triangle OFRhas a constant value.Hence, for any value ofthe acute angle0-cos B= sin A= sin(90°-B)e, there is a fixed set of values for the ratios of the varioussides of the triangle.These ratios are defined asfollowsbItan B= cot A= cot(90°B)asideoppositesine e= sin @ahypotenusecot B= tan A= tan(90°B)6side adjacentcosinee=cos0-:sec B= csc(90°B)=CSCAhypotenuseside oppositecbtangente= tan e=csc B= sec(90°-B)side adjacent= sec Ahypotenusecosecante=cScside oppositeBhypotenusesecante= sec eside adjacentside adjacentcotangent e= cot aside oppositeaOfthese sixprincipal functions,the secondthree arethere-CAbFigure2108b.Arighttriangle
NAVIGATIONAL MATHEMATICS 331 muths are usually expressed in degrees and tenths (164.7°). The system of degrees, minutes, and seconds indicated above is the sexagesimal system. In the centesimal system used chiefly in France, the circle is divided into 400 centesimal degrees (sometimes called grades) each of which is divided into 100 centesimal minutes of 100 centesimal seconds each. A radian is the angle subtended at the center of a circle by an arc having a linear length equal to the radius of the circle. A circle (360°) = 2π radians, a semicircle (180°) = π radians, a right angle (90°) = π/2 radians. The length of the arc of a circle is equal to the radius multiplied by the angle subtended in radians. 2108. Trigonometric Functions Trigonometric functions are the various proportions or ratios of the sides of a plane right triangle, defined in relation to one of the acute angles. In Figure 2108a, let θ be any acute angle. From any point R on line OA, draw a line perpendicular to OB at F. From any other point R’ on OA, draw a line perpendicular to OB at F’. Then triangles OFR and OF’R’ are similar right triangles because all their corresponding angles are equal. Since in any pair of similar triangles the ratio of any two sides of one triangle is equal to the ratio of the corresponding two sides of the other triangle, No matter where the point R is located on OA, the ratio between the lengths of any two sides in the triangle OFR has a constant value. Hence, for any value of the acute angle θ, there is a fixed set of values for the ratios of the various sides of the triangle. These ratios are defined as follows: Of these six principal functions, the second three are the reciprocals of the first three; therefore In Figure 2108b, A, B, and C are the angles of a plane right triangle, with the right angle at C. The sides are labeled a, b, c, opposite angles A, B, and C respectively. The six principal trigonometric functions of angle B are: sine θ = sin θ cosine θ = cos θ tangent θ = tan θ cosecant θ = csc θ secant θ = sec θ cotangent θ = cot θ ( ) RF,OF R’F’ OF’ - RF OR - R’F’ OR’ - and OF OR- OF’ OR’ = == = - = side opposite hypotenuse - = side adjacent hypotenuse - = side opposite side adjacent - = hypotenuse side opposite - = hypotenuse side adjacent - = side adjacent side opposite - Figure 2108a. Similar right triangles. Figure 2108b. A right triangle. sinθ 1 cscθ = - cscθ 1 sinθ = - cosθ 1 secθ = - secθ 1 cosθ = - tanθ 1 cotθ = - cotθ 1 tanθ = - B b c sin = - = = cos 90 A cos( ) ° – B B a c cos = - = = sin 90 A sin( ) ° – B B b a tan = - = = cot 90 A cot( ) ° – B B a b cot = - = = tan A 90 tan( ) ° – B B c a sec = - = = csc 90 A csc( ) ° – B B c b csc = - = = sec A 90 sec( ) ° – B
332NAVIGATIONAL MATHEMATICSSince A and B are complementary,these relationssometimescalledthenaturalfunctiontodistinguishitfromthe logarithm of the function, called the logarithmic func-showthat the sine of an angle isthe cosine of its comple-ment, the tangent of an angle is the cotangent of itstion, Numerical values of the six principal functions arecomplement, and the secant of an angle is the cosecant ofgiven at I'intervals in the table of natural trigonometricits complement.Thus,the co-function of an angleis thefunctions. Logarithms aregiven at the same intervals in an-functionof itscomplement.othertable.Sincetherelationshipsof 30°60°,and 45°righttrian-gles are as shown inFigure 2108c, certain values ofthebasicsin(90°A)=COSACOS(90°-A)=sinAfunctions canbe stated exactly,as shown in Table2108tan(90°A)=cotA2109.FunctionsInVariousQuadrantsCSC(90°-A)=secAsec(90°-A)=CSCAcot(90°A)=tanATo makethe definitions of thetrigonometric functionsThe numerical value of a trigonometric function ismoregeneral to include those angles greater than 90othe459030090045-V31Figure 2108c.Numerical relationship of sides of 300,60°,and 45°triangles.300450600Function19-1/1-sine21六-学-5cosine2六一1/5华.51 -1tangent45六-1/5I-1cotangent号一号5-2secant1元一号店21=2cosecant1Table2108.Valuesofvarious trigonometricfunctionsforangles30,45°,and600
332 NAVIGATIONAL MATHEMATICS Since A and B are complementary, these relations show that the sine of an angle is the cosine of its complement, the tangent of an angle is the cotangent of its complement, and the secant of an angle is the cosecant of its complement. Thus, the co–function of an angle is the function of its complement. The numerical value of a trigonometric function is sometimes called the natural function to distinguish it from the logarithm of the function, called the logarithmic function. Numerical values of the six principal functions are given at 1' intervals in the table of natural trigonometric functions. Logarithms are given at the same intervals in another table. Since the relationships of 30°, 60°, and 45° right triangles are as shown in Figure 2108c, certain values of the basic functions can be stated exactly, as shown in Table 2108. 2109. Functions In Various Quadrants To make the definitions of the trigonometric functions more general to include those angles greater than 90°, the sin(90° – A) =cosA cos(90° – A) =sinA tan(90° – A) =cotA csc(90° – A) =secA sec(90° – A) =cscA cot(90° – A) =tanA Figure 2108c. Numerical relationship of sides of 30°, 60°, and 45° triangles. Function 30° 45° 60° sine cosine tangent cotangent secant cosecant Table 2108. Values of various trigonometric functions for angles 30°, 45°, and 60°. 1 2 - 1 2 - 1 2 = - 2 3 2 - 1 2 = - 3 3 2 - 1 2 = - 3 1 2 - 1 2 = - 2 1 2 - 1 3 - 1 3 = - 3 1 1 - 1 = 3 1 - 3 = 3 1 - 3 = 1 1 - 1 = 1 3 - 1 3 = - 3 2 3 - 2 3 = - 3 2 1 - 2 = 2 1 - 2 = 2 1 - 2 = 2 1 - 2 = 2 3 - 2 3 = - 3
333NAVIGATIONALMATHEMATICSfunctions aredefined interms of the rectangular Cartesiansin (180°+0)=-y=-sinecoordinates of point R of Figure 2108a, due regard beingsin (360°0)=-y= sin (0) =sin givento the signofthe function.InFigure2109a,OR isas-sumed to beaumit radius.By convention the sign of OR isThe numerical value of the cosine of an angle is equalalways positive.This radius is imagined to rotate in a coun-to the projection of the unit radius on the X axis. In Figureterclockwise direction through 360ofrom the horizontal2109a,position at o°, the positive direction along the X axis. Nine-cos = +xty degrees (90°) is the positive direction along the Y axisThe angle between the original position ofthe radius and itscos (180°-0) =-x=-Cosposition at any time increases from 0° to 90° in the firstcOs (180°+0) =-X=-COs Qquadrant (I), 90°to 180° in the second quadrant (11), 1800=cos0cos (360°-0) =+x = cos (-0)to270°in thethird quadrant (11),and 270°to360°in thefourth quadrant (IV)The numerical value ofthe tangent ofan angle is equalThenumerical value of the sine of an angle is equal toto the ratio ofthe projections ofthe unit radius on the Y andthe projection of the unit radius on the Y-axis.AccordingX axes. In Figure 2109ato the definition given in article2108, the sine of angle intytythefirst quadrant ofFigure2109a is.IftheradiusOR=tan eFOR+xis equal to one, sin =+y. Since+y is equal to the projectiontan (180°8) ==tan eof the unitradius OR on theY axis, the sinefunction of an-xangle in the first quadrant defined in terms of rectangularCartesian coordinates does not contradictthe definition in-y= tan etan (180°+0)=article 2108.InFigure2109a,Xsine= +y-ytan (360°)== tan (0)=-tan e+xsin (180°0) =+y= sine+Se-AIⅡP--I+y+yI--X-axis,+O1-180°+11-6A1-360°-oIIIV1Figure2109a.Thefunctions in various quadrants,mathematical convention
NAVIGATIONAL MATHEMATICS 333 functions are defined in terms of the rectangular Cartesian coordinates of point R of Figure 2108a, due regard being given to the sign of the function. In Figure 2109a, OR is assumed to be a unit radius. By convention the sign of OR is always positive. This radius is imagined to rotate in a counterclockwise direction through 360° from the horizontal position at 0°, the positive direction along the X axis. Ninety degrees (90°) is the positive direction along the Y axis. The angle between the original position of the radius and its position at any time increases from 0° to 90° in the first quadrant (I), 90° to 180° in the second quadrant (II), 180° to 270° in the third quadrant (III), and 270° to 360° in the fourth quadrant (IV). The numerical value of the sine of an angle is equal to the projection of the unit radius on the Y–axis. According to the definition given in article 2108, the sine of angle in the first quadrant of Figure 2109a is . If the radius OR is equal to one, sin θ=+y. Since +y is equal to the projection of the unit radius OR on the Y axis, the sine function of an angle in the first quadrant defined in terms of rectangular Cartesian coordinates does not contradict the definition in article 2108. In Figure 2109a, The numerical value of the cosine of an angle is equal to the projection of the unit radius on the X axis. In Figure 2109a, The numerical value of the tangent of an angle is equal to the ratio of the projections of the unit radius on the Y and X axes. In Figure 2109a sin θ = +y sin (180°−θ) = +y = sin θ +y +OR - sin (180° +θ) = –y = –sin θ sin (360° −θ) = –y = sin (–θ) = –sin θ cos θ = +x cos (180°−θ) = –x = –cos θ cos (180°+θ) = –x = –cos θ cos (360°−θ) = +x = cos (–θ) = cos θ . tan θ = tan (180° −θ) = = –tan θ tan (180° +θ) = = tan θ tan (360° −θ) = = tan (–θ) = –tan θ . +y +x- +y –x - –y –x - –y +x- Figure 2109a. The functions in various quadrants, mathematical convention
334NAVIGATIONALMATHEMATICSThe cosecant, secant,and cotangentfunctions ofangles inThe numerical values vary by quadrant as shown above:the various quadrants are similarly determinedIIIIIIV110to+1+1 to 00 to-1-1 to 0sincsco=0o to 1ty +oo to +1+1 to 0-1 to 80cSC+1to00 to-1-1 to 00 to +1cOsoo to 11 to-8+ to +1sec+1 to+ooIcsc(180°-0)=csce+y-00 to 0tan0to+oo0o to 00to+o+ooto 0oo to 0+oo to 00 to -ocot1These relationships are shown graphically in Figurecsc(180°+0)=-csce-y2109b.2110.TrigonometricIdentities1csc(360°)=csc(-)= -csc-yA trigonometric identity is an equality involving trig-onometricfunctionsof ewhichistrueforall valuesof e1exceptthosevaluesforwhichoneofthefunctionsisnotde-secQ=+xfinedorforwhichadenominatorintheequalityisequaltozero.Thefundamental identitiesarethose identities from1which other identities can be derivedsec(180°0)=-sece-x11singAcsc=cscesine1sec(180°+)=-sece-x11cOsO=seco=cOsesece1sec(360°0)=sec(-0)= sec+x11tane=cote=tanecote+xcot e=sinecot=cose+y=tanecosesine-Xtan20 + 1= sec20sin?+cos20=1cot(180°-0)=-cote+ysin(90°-)=cosocsc(90°-0)= seco-xcos(90°-0)=singsec(90°-0)= cscocot(180°+0)=cote-ytan(90°-0)=cotecot(90°-)= tanosin(-0)= -sincsc(-0)= -csc0+xcot(360°0)=cot(-0)=-coto-ycos(-0)= cososec(-0)= secotan(-0)=-tangcot(-0)= -cotocsc(90+)=singsin(90+0)=cosThe signs ofthe functions in the four different quadrants arecos(90+0)=-singsec(90+0)=-csc0shown below:tan(90+)=-cotecot(90+)=-tano1IIIIIVsin(180°-)= singcsc(180°-0)=csco++sine and cosecant--cOs(180°-0) = -cosOsec(180°-0)=-seco++cosine and secant::tan(180°-)=-tangcot(180°-0)=-coto++1tangentandcotangent-
334 NAVIGATIONAL MATHEMATICS The cosecant, secant, and cotangent functions of angles in the various quadrants are similarly determined. The signs of the functions in the four different quadrants are shown below: The numerical values vary by quadrant as shown above: These relationships are shown graphically in Figure 2109b. 2110. Trigonometric Identities A trigonometric identity is an equality involving trigonometric functions of θ which is true for all values of θ, except those values for which one of the functions is not defined or for which a denominator in the equality is equal to zero. The fundamental identities are those identities from which other identities can be derived. I II III IV sine and cosecant + + - - cosine and secant + - - + tangent and cotangent + - + - θ 1 +y csc = - ( ° θ) 180 1 +y csc – = -= cscθ ( ° 180 +θ ) 1 -y csc = -= –cscθ ( °θ 360 – ) 1 -y csc = -= csc(–θ) θ = –csc θ 1 +x sec = - ( °θ 180 – ) 1 –x sec = -= –secθ ( ° 180 +θ ) 1 –x sec = -= –secθ ( °θ 360 – ) 1 +x sec = -= sec(–θ) θ = sec θ +x +y cot = - ( °θ 180 – ) –x +y cot = -= –cotθ ( ° 180 +θ ) –x –y cot = -= cotθ ( °θ 360 – ) +x -y cot = -= cot(–θ) θ = –cot I II III IV sin 0 to +1 +1 to 0 0 to –1 –1 to 0 csc +∞ to +1 +1 to 0 –∞ to –1 –1 to –∞ cos +1 to 0 0 to –1 –1 to 0 0 to +1 sec +1 to +∞ –∞ to –1 –1 to –∞ +∞ to +1 tan 0 to +∞ –∞ to 0 0 to +∞ –∞ to 0 cot +∞ to 0 –∞ to 0 +∞ to 0 0 to –∞ sinθ 1 cscθ = - cscθ 1 sinθ = - cosθ 1 secθ = - secθ 1 cosθ = - tanθ 1 cotθ = - cotθ 1 tanθ = - tanθ sinθ cosθ = - cotθ cosθ sinθ = - sin2θ cos2θ 1 tan2θ 1 sec2 + = + = θ sin (90° θ) θ – = cos cos(90° θ) θ – = sin tan (90° θ) θ – = cot csc(90° θ) θ – = sec sec(90° θ) θ – = csc cot(90° θ) θ – = tan sin (–θ) θ = –sin cos(–θ) θ = cos tan (–θ) θ = –tan csc (–θ) θ = –csc sec(–θ) θ = sec cot(–θ) θ = –cot sin(90+θ) θ = cos cos(90+θ) θ = –sin tan(90+θ) θ = –cot csc (90+θ) θ = sin sec (90+θ) θ = –csc cot(90+θ) θ = –tan sin (180° θ) θ – = sin cos(180° θ) θ – = –cos tan (180° θ) θ – = –tan csc(180° θ) θ – = csc sec (180° θ) θ – = –sec cot(180° θ) θ – = –cot
335NAVIGATIONALMATHEMATICSIⅢIIV1-4ooIVⅡIII9O18027036011--90270D180360ImIIV11IVIIIII-2-Z90270°0180360--a270o90180360Figure2109b.Graphic representationof valuesof trionometricfunctions invarious quadrants2111.InverseTrigonometricFunctionssin(180°+0)=-sincsc(180°+0)=-csccos(180°+0)=cos0sec(180°+0)=sec0tan(180°+0)=tanecot(180°+0)=coteAn anglehavinga giventrigonometricfunctionmaybeindicated in anyofseveral ways.Thus, sin y=x,y=arc sinx,and y= sin- x have the same meaning.The superior"1"sin(360°-)=-sinecSc(360°)=-cscis not an exponent in this case. In each case, y is “the anglecos(360°-)=cos0sec(360°-0)=sec0whose sine is x." In this case, y is the inverse sine of x. Sim-tan(360°-)=-tangcot(360°-)=-cotoilarrelationships hold for all trigonometric functions.SOLUTIONOFTRIANGLESA triangle is composed of sixparts:three angles andIn general, when any three parts are known, the other threethreesides.Theangles may bedesignatedA, B,and C,andparts can befound, unlesstheknownparts arethethreethe sides oppositethese angles as a, b,and c,respectivelyangles
NAVIGATIONAL MATHEMATICS 335 2111. Inverse Trigonometric Functions An angle having a given trigonometric function may be indicated in any of several ways. Thus, sin y = x, y = arc sin x, and y = sin–1 x have the same meaning. The superior “–1” is not an exponent in this case. In each case, y is “the angle whose sine is x.” In this case, y is the inverse sine of x. Similar relationships hold for all trigonometric functions. SOLUTION OF TRIANGLES A triangle is composed of six parts: three angles and three sides. The angles may be designated A, B, and C; and the sides opposite these angles as a, b, and c, respectively. In general, when any three parts are known, the other three parts can be found, unless the known parts are the three angles. Figure 2109b. Graphic representation of values of trionometric functions in various quadrants. sin (180°+θ) θ = –sin cos(180°+θ) θ = cos tan (180°+θ) θ = tan csc(180°+θ) θ = –csc sec(180°+θ) θ = sec cot(180°+θ) θ = cot sin (360° θ) θ – = –sin cos(360° θ) θ – = cos tan (360° θ) θ – = –tan csc(360° θ) – = –cscθ sec(360° θ) θ – = sec cot(360° θ) θ – = –cot
336NAVIGATIONALMATHEMATICSab2112. Right Plane TrianglescLaw of sines:-sinB"sinAsinCIn a right plane triangle it is only necessary to substitutenumerical values in the appropriate formulas representingLawofcosines: a? = b?+?-2bc cos Athe basic trigonometric functions and solve. Thus, ifa and bThe unknown parts of oblique plane triangles can beare known,computed by the formulas in Table 2113, among others. ByatanA=reassignment of lettersto sides and angles, theseformulasbcanbeusedto solvefor all unknownparts ofobliqueplaneB= 90°-Atriangles.c=acscASimilarly,ifcandBaregiven,A = 90°-BBa=csinAb= c cos ACa2113.ObliquePlaneTrianglesWhen solving an oblique plane triangle, it is often de-sirable todrawarough sketch of thetriangle approximatelyACbto scale,as shown inFigure2113.Thefollowinglaws arehelpful insolving suchtrianglesFigure2113.Aplane obliquetriangleKnownTofindFormulaComments2+b2-2Aa,b,cCosine lawCOSA=2bcBa,b,AbsinASinelaw.Twosolutionsifb>asinB =acA+B±C = 180°asincCSine lawcsinAasinCa, b,CAtanA=b-acoscBA+B+C=180°B = 180°-(4-C)cSinelawasincCOSA =sinAbasinca, A, BSine lawC=sinAcA+B+ C=180°C = 180°-(A+B)cSine lawasincc=sinATable2113.Formulas for solvingobliqueplanetriangles
336 NAVIGATIONAL MATHEMATICS 2112. Right Plane Triangles In a right plane triangle it is only necessary to substitute numerical values in the appropriate formulas representing the basic trigonometric functions and solve. Thus, if a and b are known, Similarly, if c and B are given, 2113. Oblique Plane Triangles When solving an oblique plane triangle, it is often desirable to draw a rough sketch of the triangle approximately to scale, as shown in Figure 2113. The following laws are helpful in solving such triangles: The unknown parts of oblique plane triangles can be computed by the formulas in Table 2113, among others. By reassignment of letters to sides and angles, these formulas can be used to solve for all unknown parts of oblique plane triangles. A a b tan = - B = 90° – A c a = csc A A = 90°–B a c = sin A b c = cos A Figure 2113. A plane oblique triangle. Law of sines: a sinA - b sinB - c sinC = = - Law of cosines: a 2 b 2 c 2 = + – 2bc cos A . Known To find Formula Comments a, b, c A Cosine law a, b, AB Sine law. Two solutions if b>a C c Sine law a, b, C A B A+B+C=180° c Sine law a, A, Bb Sine law C A + B + C=180° c Sine law Table 2113. Formulas for solving oblique plane triangles. cosA c 2 b 2 a 2 + – 2bc = - sinB b A sin a = - ABC + + 180 = ° c a C sin sinA = - tanA a C sin ba C – cos = - B = 180° – ( ) A C– cosA a C sin sinA = - c a C sin sinA = - C = 180° – ( ) A B + c a C sin sinA = -