151 Spherical Trigonometry Terms and Definitions | Mathematics Board Exam Review

INTRODUCTION

Spherical trigonometry is the branch of mathematics that deals with triangles drawn on the surface of a sphere. Unlike plane triangles where angles always sum to exactly 180 degrees, the angles of a spherical triangle sum to more than 180 degrees, and the sides are measured not in linear units but in degrees or radians as angular distances. This fundamental difference makes spherical trigonometry a subject that requires deliberate study, because many of the intuitions built from plane geometry do not transfer directly to the surface of a sphere.

In the Philippine engineering board examinations, spherical trigonometry is most prominent in the civil engineering and geodetic engineering programs, where it underpins geodesy, astronomical surveying, and terrestrial navigation. CE board exam candidates regularly encounter problems involving great circle distances between geographic points, azimuths, and latitude-longitude calculations. GeE candidates go even deeper into the celestial coordinate systems and the mathematical relationships between the observer, the celestial pole, and observed bodies. Even ECE and EE candidates occasionally encounter spherical geometry concepts in problems involving antenna coverage, satellite positioning, and coordinate transformations.

This glossary presents 151 terms from spherical trigonometry organized alphabetically. Every definition is written with the board exam context clearly in mind. You will find the core structural terms like spherical triangle and spherical excess alongside the practical navigation terms like great circle distance, nautical mile, and terrestrial triangle. The solution tools including Napier’s rules, the spherical laws of sines and cosines, and Napier’s analogies are all defined in detail with guidance on when to apply each. The coordinate system terms covering latitude, longitude, declination, and hour angle are explained in relation to the spherical triangles they help form.

Study this glossary as part of your systematic review of spherical trigonometry, not in isolation. Connect each term to a formula, connect each formula to a problem type, and connect each problem type to the specific examination context where it is most likely to appear. Spherical trigonometry problems on the board exam reward candidates who understand the geometry of the sphere deeply enough to set up the correct triangle before applying any formula. That setup skill begins with knowing your vocabulary cold.

The 151 Spherical Trigonometry Terms and Definitions 

1. Altitude (Celestial)

The angular elevation of a celestial body above the observer’s horizon, measured along the vertical circle passing through the body. Altitude ranges from 0 degrees at the horizon to 90 degrees at the zenith. It is one of the two horizontal coordinates (the other being azimuth) that describe the position of a celestial body as seen from a specific location on Earth. In the astronomical triangle, the altitude appears as one of the sides and is a primary observable quantity in celestial navigation.

2. Altitude Circle

A small circle on the celestial sphere connecting all points at the same altitude above the observer’s horizon. All celestial bodies on an altitude circle appear at the same elevation angle from the observer’s location. In navigation, altitude circles are used to define lines of position (circles of equal altitude). When two altitude circles from simultaneous observations of different celestial bodies are combined, their intersection gives the observer’s position. This is the geometric basis of the intercept method of celestial navigation.

3. Angle Sum of a Spherical Triangle

The sum of the interior angles of any spherical triangle always exceeds 180 degrees and is always less than 540 degrees. This is the most fundamental property distinguishing spherical triangles from plane triangles. The excess above 180 degrees is called the spherical excess. As the spherical triangle becomes very small relative to the sphere, its angle sum approaches 180 degrees and it behaves more like a plane triangle. The angle sum property is directly related to the area of the spherical triangle through Girard’s theorem.

4. Angular Distance

The angle subtended at the center of a sphere by two points on its surface, measured in degrees or radians. On Earth, the angular distance between two geographic points is the central angle between them as seen from Earth’s center. This angular distance, when multiplied by the Earth’s radius, gives the great circle arc length in linear units. In spherical trigonometry, the sides of a spherical triangle are angular distances, not linear lengths.

5. Angular Radius

The angular distance from the center of a circle on a sphere to any point on the circumference of that circle. For a small circle on a sphere, the angular radius is the angle between the axis of the circle and any radius drawn to the circle’s edge. The angular radius of a parallel of latitude equals 90 degrees minus the latitude. This concept connects the geometry of small circles to the angular coordinate system used in spherical trigonometry.

6. Angular Separation

The angle between the directions from an observer to two different celestial bodies, measured along the great circle arc connecting their positions on the celestial sphere. Angular separation is computed using the spherical law of cosines: cos d = sin δ₁ sin δ₂ + cos δ₁ cos δ₂ cos(α₁ − α₂), where δ is declination and α is right ascension. In engineering applications, angular separation determines whether two satellites are close enough to cause interference, and whether two stars are close enough for a specific telescope or antenna to resolve them.

7. Antipodal Point

The point on a sphere directly opposite a given point, located at the same distance from the center but in exactly the opposite direction. The antipodal point of any location on Earth is the geographic point reached by passing through Earth’s center. For a point at latitude φ and longitude λ, its antipodal point is at latitude −φ and longitude λ ± 180°. Every great circle through a point also passes through its antipodal point. Understanding antipodal points is important for analyzing great circles and for certain satellite coverage problems.

8. Aphelion

The point in Earth’s elliptical orbit around the Sun where Earth is farthest from the Sun. While primarily an astronomical term, aphelion appears in spherical trigonometry problems involving the celestial sphere because the position of the Sun on the ecliptic changes as Earth moves from perihelion to aphelion. Problems involving the declination and right ascension of the Sun at different times of year require awareness of Earth’s orbital position.

9. Arc of a Great Circle

A portion of a great circle on a sphere, used as the side of a spherical triangle. The length of a great circle arc is proportional to the central angle it subtends. In spherical trigonometry, the sides a, b, and c of a spherical triangle are expressed as angular measures (the central angles in degrees or radians), not as linear lengths. This is why the sides and angles of a spherical triangle are all measured in the same units.

10. Astronomical Coordinates

The angular values that specify the position of a celestial body on the celestial sphere within a chosen coordinate system. The three principal systems are: the horizontal system (altitude and azimuth), the equatorial system (declination and hour angle, or declination and right ascension), and the ecliptic system (ecliptic latitude and ecliptic longitude). Converting between these coordinate systems requires solving spherical triangles using the laws of spherical trigonometry. The astronomical triangle connects the horizontal and equatorial systems.

11. Astronomical Triangle

The spherical triangle formed on the celestial sphere by three vertices: the zenith of the observer, the north celestial pole, and the position of a celestial body. The three sides of the astronomical triangle are the co-altitude (equal to 90 degrees minus the altitude), the co-declination or polar distance (equal to 90 degrees minus the declination), and the co-latitude (equal to 90 degrees minus the latitude). The three angles are the azimuth, the hour angle, and the parallactic angle. This triangle is the central computational tool for celestial navigation and astronomical observation problems in the board exam.

12. Azimuth

The horizontal direction of a celestial body or terrestrial point, measured as an angle in the horizontal plane, usually from true north in a clockwise direction, ranging from 0 to 360 degrees. In the astronomical triangle, the azimuth is the angle at the zenith vertex, measured from the north toward the east. In surveying and geodesy problems, azimuth is one of the primary unknowns solved for using the spherical law of sines or cosines applied to the astronomical or terrestrial triangle.

13. Bearing

The direction of a point or line on Earth’s surface relative to true north or geographic north. Bearings are expressed either as compass bearings (for example, N 45° E) or as azimuths (measured clockwise from north, 0 to 360 degrees). In spherical trigonometry navigation problems, the bearing of one geographic point from another is related to the azimuth of the great circle arc connecting them at the point of departure. Converting between bearing and azimuth formats and applying spherical trigonometry to find bearings over long distances is a tested board exam skill.

14. Birectangular Spherical Triangle

A spherical triangle that contains exactly two right angles. In a birectangular spherical triangle, the two right angles determine that the two sides adjacent to the third angle each measure 90 degrees. The third angle equals the angular measure of the side opposite it. Birectangular spherical triangles are special cases used in theoretical developments of spherical trigonometry formulas and appear in problems illustrating the differences between spherical and plane geometry.

15. Celestial Equator

The great circle on the celestial sphere located directly above Earth’s geographic equator. The celestial equator divides the celestial sphere into the northern and southern celestial hemispheres. Declination is measured north or south of the celestial equator in degrees. The Sun crosses the celestial equator at the vernal and autumnal equinoxes, when day and night are of equal length everywhere on Earth. In spherical trigonometry problems involving celestial positions, the celestial equator is the primary reference plane for the equatorial coordinate system.

16. Celestial Horizon

The great circle on the celestial sphere located 90 degrees from the zenith of the observer. It is the boundary between the visible and invisible portions of the celestial sphere from a given location. Celestial bodies at the horizon have an altitude of exactly 0 degrees. The celestial horizon is the reference plane for the horizontal coordinate system, in which altitude and azimuth are measured. It differs from the apparent horizon seen by an observer at sea level because the celestial horizon is a geometric construct based on the plumb line through the observer.

17. Celestial Meridian

The great circle on the celestial sphere that passes through the zenith of the observer and the celestial poles. The celestial meridian divides the celestial sphere into eastern and western halves as seen from the observer. A celestial body reaches its highest point in the sky (upper transit) when it crosses the celestial meridian. Hour angles are measured from the celestial meridian, increasing westward. This concept is central to problems involving local sidereal time and the transit method of astronomical observation.

18. Celestial Pole

The point on the celestial sphere directly above the geographic north or south pole of Earth. The north celestial pole is located near the star Polaris and serves as the fixed center around which all celestial bodies appear to rotate due to Earth’s daily rotation. In the astronomical triangle, the celestial pole is one of the three vertices. The altitude of the north celestial pole above the horizon equals the observer’s geographic latitude, a fundamental relationship used in latitude determination from star observations.

19. Celestial Sphere

The imaginary sphere of infinite radius centered on the observer (or on Earth’s center) onto which all celestial bodies appear to be projected. The celestial sphere is the geometric framework for all astronomical and navigational coordinate systems. The equatorial, horizontal, and ecliptic coordinate systems are all defined on the celestial sphere using great circles and angular measurements. Understanding the geometry of the celestial sphere is the conceptual foundation for all spherical trigonometry problems in astronomical and navigational contexts.

20. Circumpolar Star

A star that never sets below the observer’s horizon because its angular distance from the celestial pole is less than the observer’s latitude. A star is circumpolar if its declination satisfies: δ > 90° − φ (for northern hemisphere observers), where δ is the star’s declination and φ is the observer’s latitude. Circumpolar stars can always be observed on any clear night regardless of the season. Computing the minimum altitude of a circumpolar star (at lower transit) and its maximum altitude (at upper transit) involves straightforward spherical triangle relationships.

21. Co-altitude

The complement of the altitude of a celestial body, equal to 90 degrees minus the altitude. The co-altitude is the angular distance from the zenith to the celestial body and is one of the three sides of the astronomical triangle. It is also called the zenith distance. In the formulation of the astronomical triangle, the three sides are expressed as co-altitude, co-declination, and co-latitude, making it convenient to apply Napier’s rules and the spherical law of cosines consistently.

22. Co-declination

The complement of the declination of a celestial body, equal to 90 degrees minus the declination. The co-declination is also called the polar distance because it represents the angular distance from the celestial pole to the celestial body. It is one of the three sides of the astronomical triangle. When the declination of a body is given, the co-declination is computed directly and inserted into the spherical triangle as the side connecting the pole and the body.

23. Co-latitude

The complement of the geographic latitude of an observer’s location on Earth, equal to 90 degrees minus the latitude. In the astronomical triangle, the co-latitude is the side connecting the zenith and the celestial pole. Since the altitude of the celestial pole equals the observer’s latitude, the co-latitude represents the angular distance from the zenith down to the pole’s position. This side is a fixed quantity for a given observation location and is one of the three sides of the astronomical triangle.

24. Colunar Triangle

A spherical triangle formed by extending two sides of a given spherical triangle to form a new triangle using the supplementary arcs. Colunar triangles are used in proofs and derivations in spherical trigonometry to establish relationships between angles and sides of the original triangle. Understanding colunar triangles deepens comprehension of why certain spherical identities hold and how the supplementary angle properties of the sphere interact with the triangle’s geometry.

25. Colure

A great circle on the celestial sphere that passes through both celestial poles. The two primary colures are the equinoctial colure (passing through the vernal and autumnal equinoxes) and the solstitial colure (passing through the summer and winter solstices). The equinoctial colure is the zero-point reference for measuring right ascension. While colures are primarily used in advanced astronomical problems, knowing their definition helps you understand the geometry of the equatorial coordinate system.

26. Complement of an Arc

On a sphere, the complement of an arc a is the arc (90° − a). Complements of arcs appear extensively in Napier’s circle formulation, where the hypotenuse and the two oblique angles of a right spherical triangle are replaced by their complements. Working with complements allows Napier’s rules to be stated compactly in terms of adjacent and opposite circular parts. Correctly identifying complements is the key step in applying Napier’s rules without error.

27. Conical Projection

A method of mapping the surface of a sphere onto a cone that is then unrolled into a flat surface. While primarily a cartographic concept, conical projections are grounded in spherical trigonometry because the angular relationships between geographic points must be preserved or systematically distorted in a known way. Understanding the connection between spherical coordinates and map projections is part of the geodetic engineering curriculum and appears in some advanced board exam problems.

28. Convergence of Meridians

The angular difference between grid north and true north at a given point, arising from the fact that meridians converge toward the geographic poles on a sphere but appear as parallel vertical lines on a flat map. In geodetic surveying, the convergence of meridians must be accounted for when converting between geographic bearings and grid bearings. It is computed using spherical trigonometry formulas involving latitude and the difference in longitude between the point and the central meridian.

29. Cotangent Formula (Spherical)

Also called the four-parts formula or the analogue formula. For four consecutive parts of a spherical triangle (two angles and the side between them, or two sides and the angle between them), the formula is: cot a sin b = cos b cos C + sin C cot A, where a and b are sides and A and C are angles. This formula is used when neither the spherical law of sines nor the standard spherical law of cosines directly gives the required unknown. It is particularly useful when the given and required elements are adjacent in the triangle.

30. Declination

The angular distance of a celestial body north or south of the celestial equator, measured along the hour circle through the body. Declination is the celestial analog of geographic latitude and ranges from −90 degrees at the south celestial pole to +90 degrees at the north celestial pole. In the astronomical triangle, the declination of the observed body determines the co-declination (polar distance), which is one of the three sides. Navigation problems that involve computing altitude or azimuth from known latitude, declination, and hour angle require setting up the astronomical triangle with declination as one of the given quantities.

31. Delambre’s Analogies

An alternative name for Napier’s analogies in spherical trigonometry, named after the French astronomer Jean Baptiste Joseph Delambre. These four formulas relate the half-sums and half-differences of the sides and angles of a spherical triangle in a way analogous to Mollweide’s equations in plane trigonometry. In some engineering Mathematics references, these formulas are listed under Delambre rather than Napier. Both names refer to the same set of formulas, and knowing the formulas themselves is what matters for the board exam.

32. Departure (Navigation)

The distance traveled in the east-west direction along a parallel of latitude. For a ship traveling from one longitude to another at a fixed latitude φ, the departure is computed as: departure = R cos φ × |Δλ|, where Δλ is the change in longitude in radians. Departure is used in parallel sailing and in the construction of the traverse table for plane sailing calculations. Understanding departure as a trigonometric quantity connecting longitude difference and linear east-west distance is important for navigation problems on the CE and GeE board exams.

33. Difference in Latitude

The angular change in latitude between two geographic points, computed as φ₂ − φ₁. It is used in dead reckoning navigation calculations and in the formulation of sailing problems. The linear distance corresponding to the difference in latitude is independent of longitude and is computed as: distance = R × |Δφ| (for angular difference in radians) or as nautical miles by converting degrees to arc minutes. Difference in latitude is a building block in the solution of mid-latitude sailing and Mercator sailing problems.

34. Difference in Longitude

The angular difference between the longitudes of two geographic points, taken as the smaller of the two possible values (not exceeding 180 degrees) unless a specific direction (east or west) is required. In the terrestrial triangle, the difference in longitude is the angle at the pole vertex. It is the direct angular input to the great circle distance formula. For navigation problems involving departure and mid-latitude sailing, the difference in longitude combined with the cosine of the mid-latitude gives the departure.

35. Dihedral Angle

The angle between two planes, measured as the angle between two lines, one in each plane, that are perpendicular to the line of intersection of the planes. In spherical trigonometry, the angles of a spherical triangle are dihedral angles between the planes of the great circles that form its sides. This is why the angles of a spherical triangle can individually exceed 90 degrees and why their sum can reach nearly 540 degrees. Understanding that spherical triangle angles are dihedral angles clarifies why they behave so differently from plane triangle angles.

36. Diurnal Motion

The apparent daily east-to-west motion of all celestial bodies across the sky due to Earth’s rotation from west to east. Diurnal motion causes celestial bodies to rise in the east, reach their highest point at upper transit on the observer’s meridian, and set in the west. The diurnal circle of a celestial body is the small circle parallel to the celestial equator along which the body appears to move during one day. The angular rate of diurnal motion is 15 degrees per hour for all celestial bodies, corresponding to one full revolution (360 degrees) in 24 hours.

37. Ecliptic

The apparent path of the Sun across the celestial sphere over the course of one year, caused by Earth’s orbital revolution. The ecliptic is a great circle inclined at about 23.5 degrees to the celestial equator, an angle called the obliquity of the ecliptic. The ecliptic intersects the celestial equator at the vernal and autumnal equinoxes. In spherical trigonometry problems involving the Sun’s position, the ecliptic coordinate system is used alongside the equatorial system, requiring spherical triangle calculations to convert between them.

38. Ecliptic Coordinates

The coordinate system that uses the ecliptic as its reference plane and measures positions by ecliptic longitude (measured eastward from the vernal equinox along the ecliptic) and ecliptic latitude (measured perpendicular to the ecliptic). The conversion between ecliptic coordinates and equatorial coordinates requires spherical triangle calculations involving the obliquity of the ecliptic as a known side. Problems involving the positions of planets and the Moon in relation to the Sun sometimes use ecliptic coordinates.

39. Equator (Celestial)

See Celestial Equator. This shortened form appears frequently in board exam problems and references. It is the great circle on the celestial sphere directly above Earth’s geographic equator and is the reference plane for measuring declination.

40. Equatorial Bulge

The slight flattening of Earth at the poles and corresponding bulging at the equator due to Earth’s rotation. The equatorial radius is about 21 km greater than the polar radius. In precise geodetic calculations, the equatorial bulge means that the distance between two points at the same latitude but different longitudes is slightly different from what a perfectly spherical Earth model would predict. For engineering board exam problems, this effect is ignored and Earth is treated as a perfect sphere.

41. Equatorial Coordinate System

The coordinate system on the celestial sphere that uses the celestial equator as its reference plane and measures positions by right ascension (analogous to longitude) and declination (analogous to latitude). This system is fixed with respect to the stars and does not rotate with Earth’s daily motion, making it convenient for star catalogs and long-term astronomical calculations. Problems involving the positions of stars and the transformation between equatorial and horizontal coordinates require spherical trigonometry applied to the astronomical triangle.

42. Equinox

Either of the two points where the ecliptic intersects the celestial equator. At the vernal equinox (around March 21), the Sun crosses the celestial equator moving northward. At the autumnal equinox (around September 23), it crosses moving southward. On both equinox dates, the length of day and night are approximately equal everywhere on Earth. The vernal equinox serves as the zero-point for measuring right ascension. In spherical trigonometry problems, equinox dates are used as reference points for computing the Sun’s declination.

43. Excess (Spherical)

See Spherical Excess. The spherical excess E is the amount by which the sum of the angles of a spherical triangle exceeds 180 degrees. It equals A + B + C − 180°. The area of a spherical triangle is proportional to its spherical excess: Area = E × R², where E is in radians and R is the sphere’s radius. This relationship, called Girard’s theorem, is the fundamental formula connecting the geometry of spherical triangles to their areas.

44. Four-Parts Formula

See Cotangent Formula (Spherical). The four-parts formula is the most common name for the cotangent rule in spherical trigonometry as it appears in engineering Mathematics review books. It involves four consecutive elements of a spherical triangle and is used when the standard law of sines or law of cosines does not conveniently connect the given and required parts.

45. Geocentric Latitude

The angle between the equatorial plane and the line connecting a point on Earth’s surface to Earth’s center. Because Earth is an oblate spheroid rather than a perfect sphere, geocentric latitude differs slightly from geographic (geodetic) latitude except at the equator and poles. In spherical trigonometry problems that treat Earth as a perfect sphere, geocentric and geographic latitude are assumed to be equal. Understanding this distinction is important for geodetic engineering candidates who deal with the precise shape of Earth.

46. Geodetic Latitude

The angle between the equatorial plane and the normal to the reference ellipsoid at a given point on Earth’s surface. Geodetic latitude is the standard meaning of latitude used in GPS coordinates and in precise geodetic survey work. It differs from geocentric latitude by a small correction that depends on the flattening of the Earth ellipsoid. For spherical approximation problems in engineering board exams, geodetic and geocentric latitude are treated as equal.

47. Geodesic

The shortest path between two points on the surface of a sphere. On a sphere, all geodesics are arcs of great circles. The geodesic distance between two points is the great circle arc length connecting them. In navigation, the geodesic route is the actual shortest flight path between two cities, following a great circle. The great circle distance formula derived from the spherical law of cosines gives the angular geodesic distance, which is then multiplied by the sphere’s radius to obtain the linear distance.

48. Geographic Latitude

The angle between the equatorial plane and the normal to Earth’s surface at a given point, measured in degrees north or south of the equator. Geographic latitude is the standard meaning of “latitude” in most engineering problems. It is one of the two primary geographic coordinates, the other being longitude. In spherical trigonometry navigation problems, the latitude of each geographic point determines the co-latitude, which is one of the sides of the terrestrial or astronomical triangle.

49. Geographic Pole

Either of the two points on Earth’s surface where the Earth’s rotational axis meets the surface. The north geographic pole is at 90 degrees north latitude and the south geographic pole is at 90 degrees south latitude. The celestial poles are located directly above the geographic poles. In terrestrial navigation problems using spherical trigonometry, the geographic poles are vertices of the meridian great circles that form the framework for the latitude-longitude coordinate system.

50. Girard’s Theorem

The theorem stating that the area of a spherical triangle on a sphere of radius R is given by Area = (E/180°) × πR², where E is the spherical excess in degrees, or equivalently Area = E × R² when E is in radians. Named after the French mathematician Albert Girard, this theorem is the fundamental area formula for spherical triangles. Board exam problems that ask for the area of a spherical triangle require direct application of Girard’s theorem after computing the spherical excess from the triangle’s angles.

51. Gnomonic Projection

A map projection in which great circles appear as straight lines. The gnomonic projection projects the sphere from its center onto a tangent plane. Because all great circles on a sphere appear as straight lines in the gnomonic projection, this projection is particularly useful for great circle navigation planning. A navigator can draw a straight line between two points on a gnomonic chart to determine the great circle route, then transfer waypoints to a Mercator chart for rhumb line segment plotting.

52. Great Circle

The largest possible circle that can be drawn on a sphere, formed by the intersection of the sphere with a plane that passes through the center of the sphere. Every great circle divides the sphere into two equal hemispheres. The equator and all meridians (lines of longitude) are great circles on Earth. Great circles are the geodesics of a sphere. Geodesics is the paths of shortest distance. In spherical trigonometry, the sides of every spherical triangle are arcs of great circles. This is not merely a convention; it is required because only great circle arcs define the angular distance framework that makes spherical trigonometry consistent.

53. Great Circle Distance

The angular distance between two points on a sphere, measured along the great circle arc connecting them. For two points with latitudes φ₁ and φ₂ and a difference in longitude of Δλ, the angular great circle distance d is given by: cos d = sin φ₁ sin φ₂ + cos φ₁ cos φ₂ cos Δλ. Multiplying the angular distance in radians by Earth’s radius gives the linear great circle distance. In navigation problems on the board exam, the great circle distance between two cities or geographic points is the most commonly requested quantity.

54. Great Circle Sailing

The navigation method that follows a great circle route between two geographic points to minimize the distance traveled. In practice, the great circle route is approximated by a series of straight-line (rhumb line) segments. Computing the initial course and total distance of a great circle sailing route requires solving a spherical triangle using the spherical law of cosines and the spherical law of sines. This is one of the most practically relevant applications of spherical trigonometry in naval and aeronautical engineering.

55. Greenwich Hour Angle

The hour angle of a celestial body measured from the Greenwich meridian (0 degrees longitude) rather than from the observer’s local meridian. The Greenwich hour angle (GHA) increases westward and is used in nautical almanacs to give the position of the Sun, Moon, planets, and stars throughout the day. The local hour angle at any location equals the Greenwich hour angle plus the east longitude (or minus the west longitude) of the observer. GHA and local hour angle together connect the rotating Earth frame to the fixed celestial coordinate system.

56. Greenwich Meridian

The prime meridian, the meridian passing through the Royal Observatory in Greenwich, England, defined as 0 degrees longitude. All other longitudes are measured east or west from the Greenwich meridian up to 180 degrees. In navigation problems, the difference in longitude between two geographic points is a direct input to the great circle distance formula. Time zones are also defined relative to the Greenwich meridian, with each 15 degrees of longitude corresponding to one hour of time difference.

57. Haversine Formula

A specific formula used in navigation to compute the great circle distance between two points on a sphere given their latitudes and longitudes: hav(d) = hav(φ₂ − φ₁) + cos φ₁ cos φ₂ × hav(Δλ), where hav(θ) = sin²(θ/2). This formula is numerically stable for small distances, unlike the direct application of the cosine rule which can suffer from rounding errors. It is used in GPS systems, aviation navigation software, and marine navigation and appears in advanced navigation problems in the engineering board exam.

58. Horizontal Coordinate System

The coordinate system that describes the position of a celestial body relative to the observer’s horizon and the direction of north. The two coordinates are altitude (angular elevation above the horizon) and azimuth (horizontal direction from north). This system is observer-dependent and time-dependent because it rotates with the celestial sphere as seen from a fixed point on Earth. Problems using the horizontal coordinate system are solved through the astronomical triangle, which connects horizontal coordinates to the equatorial coordinates of the celestial body.

59. Hour Angle

The angular distance of a celestial body west of the observer’s meridian, measured along the celestial equator from the meridian to the hour circle of the body. The hour angle increases at a rate of 15 degrees per hour as Earth rotates eastward. At upper transit, the hour angle is zero. The hour angle is one of the angles in the astronomical triangle and connects the equatorial coordinate system (declination and right ascension) to the observer’s local frame of reference. In board exam problems, hour angle is used with latitude and declination to compute altitude and azimuth.

60. Hour Circle

A great circle on the celestial sphere that passes through both celestial poles and through a given celestial body. Hour circles are the celestial analogs of meridians on Earth. Declination is measured along the hour circle from the celestial equator to the body. The hour angle of a body is the angle between the observer’s celestial meridian and the hour circle of the body. In the astronomical triangle, the side connecting the celestial pole and the celestial body lies along the hour circle.

61. Hypsometric Formula

A formula relating atmospheric pressure to altitude based on the assumption of a uniform atmospheric model. While its derivation involves logarithms rather than spherical trigonometry directly, altitude determination from barometric pressure readings is complementary to celestial altitude determination in navigation. In the context of spherical trigonometry, altitude here refers to the celestial quantity (angular elevation above the horizon), not the elevation above sea level, and the two should not be confused in board exam problems.

62. International Date Line

The imaginary line at approximately 180 degrees longitude where the calendar date changes by one day. When crossing the International Date Line from west to east, one day is subtracted. When crossing from east to west, one day is added. While primarily a timekeeping concept, the International Date Line is relevant to long-range navigation problems involving great circle routes that cross the 180-degree meridian. In these problems, longitude differences exceeding 180 degrees must be handled by subtracting from 360 degrees to get the shorter arc.

63. Isogonic Lines

Lines on a map connecting points of equal magnetic declination (the angular difference between true north and magnetic north). Magnetic declination varies with geographic location and changes slowly over time due to the movement of Earth’s magnetic poles. In engineering surveying problems, the magnetic declination at a specific location must be added to or subtracted from magnetic bearings to obtain true bearings. This correction is applied before using the true bearing in spherical trigonometry navigation calculations.

64. Isosceles Spherical Triangle

A spherical triangle in which two sides are equal in angular measure. As with plane isosceles triangles, the angles opposite the equal sides are also equal. Isosceles spherical triangles appear in navigation problems with symmetric geometric configurations and in problems involving regular polyhedra inscribed in spheres. The equal angle property reduces the number of unknowns in the triangle and simplifies the application of the spherical law of sines.

65. Latitude

The angular distance of a point on Earth’s surface north or south of the equator, measured in degrees from 0 at the equator to 90 at the poles. Latitude is one of the two fundamental geographic coordinates used to specify any location on Earth. In spherical trigonometry, the latitude of a geographic point determines the co-latitude, which serves as one of the sides of the terrestrial or astronomical triangle. The Philippines lies between approximately 4 degrees and 21 degrees north latitude.

66. Latitude by Meridian Altitude

A method of determining geographic latitude from a celestial observation by measuring the altitude of a celestial body when it is on the observer’s meridian (at upper transit). The formula is: latitude = declination ± (90° − altitude). The sign convention depends on whether the body transits north or south of the zenith and whether the declination is north or south. This is one of the simplest methods of celestial navigation and requires only the measurement of the noon altitude of the Sun or the transit altitude of a known star.

67. Local Hour Angle

The hour angle of a celestial body as measured from the observer’s local meridian. It is the specific value of the hour angle for a given observation location and time. Local hour angle is distinguished from Greenwich hour angle, which is measured from the Greenwich meridian. The local hour angle is computed by adding the observer’s longitude (in degrees, treated as an hour angle) to the Greenwich hour angle of the body. It is one of the three angles in the astronomical triangle.

68. Local Sidereal Time

The hour angle of the vernal equinox as measured from the observer’s meridian. Local sidereal time equals the right ascension of any celestial body that is currently crossing the observer’s meridian (at upper transit). It provides the connection between the fixed equatorial coordinate system (right ascension and declination) and the rotating local frame of reference. In board exam problems, local sidereal time is used to compute the local hour angle from the right ascension of a celestial body.

69. Longitude

The angular distance of a point on Earth’s surface east or west of the prime meridian, measured in degrees from 0 to 180 east or west. Longitude is the second fundamental geographic coordinate. In spherical trigonometry navigation problems, the difference in longitude between two geographic points, called the departure in longitude or delta-lambda, is a direct input to the great circle distance formula. The Philippines lies between approximately 116 degrees and 127 degrees east longitude.

70. Longitude by Time Sight

A method of determining geographic longitude from a celestial observation by measuring the altitude of a celestial body and the exact time of observation, then computing the expected local hour angle from the astronomical triangle and comparing it to the Greenwich hour angle from the nautical almanac. The difference gives the longitude. This method requires an accurate chronometer. It is more computationally involved than latitude determination because it requires solving the full astronomical triangle, but it is one of the traditional methods of celestial longitude determination tested in advanced navigation and geodesy board exam problems.

151. Zone Time

The standard time used within a time zone, defined as the solar time at the central meridian of the zone. Each standard time zone spans 15 degrees of longitude (one hour). Zone time differs from local apparent solar time by the longitude correction (4 minutes per degree of longitude difference from the zone meridian) and by the equation of time (a small correction for the eccentricity of Earth’s orbit and the obliquity of the ecliptic). In navigation problems, converting between zone time and Greenwich Mean Time requires knowing the time zone number, which equals the longitude of the central meridian divided by 15.

CONCLUSION

Spherical trigonometry is a compact but demanding subject, and your study effort should be focused on the areas most directly tested on the Philippine engineering Mathematics board exam. For any program, the foundation is understanding the spherical triangle itself: its six parts, the constraints on those parts, and the key fact that the angle sum exceeds 180 degrees. Build from there to the spherical laws of sines and cosines, which are the primary tools for solving oblique spherical triangles. For the CE and GeE board exams, the great circle distance formula and the terrestrial triangle are the most frequently tested applications, and you should be able to set up and solve these problems fluently using the spherical law of cosines for sides.

For right spherical triangles, Napier’s rules are the most efficient solution tool, and you should practice until applying them is automatic. The key to using Napier’s rules correctly is correctly identifying the five circular parts and remembering which are complements. The error most examinees make is forgetting to replace the hypotenuse and the two oblique angles with their complements when constructing Napier’s circle. Practice multiple right spherical triangle configurations until the process feels natural. The astronomical triangle is the major application context for right spherical triangle techniques, and understanding its three vertices (zenith, pole, and celestial body) along with its three sides (co-altitude, co-latitude, and polar distance) and three angles (azimuth, hour angle, and parallactic angle) gives you the complete framework for all celestial navigation problems.

Finally, keep the coordinate systems clear in your mind. The horizontal system (altitude and azimuth), the equatorial system (declination and hour angle, or declination and right ascension), and the geographic system (latitude and longitude) are all connected through spherical triangles. The board exam may ask you to move between these systems by solving an astronomical or terrestrial triangle. The 151 terms in this glossary cover the full vocabulary you need to navigate these problems with confidence. Know your terms, understand the geometry behind them, and practice the calculations until the setup and solution process is second nature.

For practice problems on all these topics, head over to our Spherical Trigonometry Problems and Solutions section here on PinoyBix. Hundreds of solved exam-type questions, complete with step by step solutions, organized by topic so you can drill exactly what you need to work on.