Spherical UNIT 6 SPHERICAL TRIGONOMETRY Trigonometry Structure 6.1 Introduction Objectives 6.2 Definitions 6.3 Formulae for a Spherical Triangle 6.3.1 Cosine Formula 6.3.2 Sine Formula 6.3.3 Supplemental Cosine Formula 6.4 Latitudes and Longitudes 6.5 The Haversine Formula 6.6 Cotangent Formula or Four Consecutive Parts Formula 6.7 Right-angled Spherical Triangles 6.7.1 Formulae for a Right-angled Triangle 6.7.2 Napier’s Rules 6.8 Quadrantal Spherical Triangles 6.9 Polar Triangle 6.10 Summary 6.11 Answers to SAQs 6.1 INTRODUCTION Originally trigonometry was conceived only to deal with the measurement of triangles but this branch of mathematics later included the treatment of all angular measurements and calculations. Plane trigonometry deals with such relationships when all the angles and the lines lie on a plane. The spherical trigonometry on the other hand, deals with these relationships when the angles and the lines lie on the surface of the sphere. Spherical trigonometry is thus concerned with the sphere and with the spherical triangles. We have already seen in the earlier units that a straight line drawn from the centre of the sphere to its surface is called its radius and the straight line drawn through the centre and terminated both ways on the surface is called its diameter. Objectives After studying this unit, you should be able to • distinguish between plane trigonometry and spherical trigonometry, • define and explain, great and small circles, axis and poles of great circles, • define spherical angle and also the angle of intersection between two great circles, • discuss and derive cosine, sine and supplemental cosine formulae and the limitations of their application, • explain the Haversine formula, discuss its advantage over sine and cosine formulae as also its application to problems in spherical triangles, • define right-angled spherical triangle and discuss its properties, 127 Mathematics-I • define quadrantal spherical triangles and their use in the solution of problems, • explain Nepier’s rule for right-angled triangles and quadrantal triangles and its working rule, and • explain polar triangles and their relationship with the primitive spherical triangles and their use in the solution of problem in spherical trigonometry. 6.2 DEFINITIONS Great and Small Circles When a plane cuts a sphere the section thus formed is called a circle and when the plane passes through the centre of the sphere the circle is defined as a great circle. If the plane does not pass through the centre of the sphere, the circle of the section is known as a small circle. A great circles divides the sphere into two identical parts, each called a hemisphere. Axis and Poles The extremities of the diameter perpendicular to a section of the sphere are called the poles of that section while the diameter itself is known as the axis of that section. The pole of a circle is a point on the surface of the sphere which is equidistant from all points on the circle. P is also the pole of the great circle LL′, and all great circles as is clear, have two poles. The arc PA of the great circle PAP′ is called the spherical radius of the circle QQ′ (Figure 6.1). P Q r O Q’ A L C L’ B P’ Figure 6.1 The Diameter and the Lune The intersection of the planes of two great circles is a diameter of the sphere; and these two great circles cut off on the surface of a sphere two pairs of congruent areas, which are known as Lunes. Thus a lune is a portion of the surface of a sphere enclosed by two great circles. For instance in Figure 6.1, PQLP′AP is a lune and ∠ QPA is the angle of the lune. Angle between Two Great Circles (a) The angle between the two great circles is the angle between the tangents drawn to the two great circles at their point of intersection. Thus ∠ TAT1 is the angle between the two great circles at the point A, which is the angle of their intersection. Clearly ∠ TAT1 is the angle between the planes, i.e. ∠ EOB (Figure 6.2). Also ∠ EOB = ∠ LOD. This angle is defined as spherical angle. 128 D L Spherical Trigonometry θ O E θ B T A T Figure 6.2 : Spherical Angle 1 (b) The angle between any two great circles is measured by the arc they subtend on the great circle to which they are perpendicular, i.e. the angle between the planes QPQ′ and QAQ′ is equal to the arc AP of the great circle PAP′ to which QOQ′ is perpendicular (Figure 6.3). Q P O P ’ A Q ’ Figure 6.3 (c) If the two great circles have their poles at the points L and L′, the angle of intersection of the two great circles will be equal to the angle subtended at the centre of the sphere by the arc joining the poles L and L′ of the great circles. In other words this is the angle subtended at O by the arc of a great circle joining L and L′. This is also equal to the inclination of the planes of the two great circles AA′ and BB′ (Figure 6.4). L L’ B A’ A O B’ Figure 6.4 Length of Small Circle Arc Consider a small circle arc BD of the circle BDB′ with its centre at O and pole L. Let the great circle arcs LD and LB meet the great circle AEA′, having its pole at L, in the points E and A (Figure 6.5). Then the arc BD will be given by arc BD = arc AE cos AB meaning thereby that the length of the arc BD is equal to the arc AE multiplied by the cosine of the angular distance of arc BD from the arc AE. 129 L B’ O B Mathematics-I D A’ C A E Figure 6.5 Spherical Triangle (Definition and Properties) A spherical triangle is defined as a triangle on the surface of a sphere contained by the arcs of three great circles. In the Figure 6.6 there are three great circles ABA′B′, ACA′C′, BCB′C′ of the sphere with centre O and BAC is one of the spherical triangles formed by them. It must be noted that the sides of the spherical triangles are formed only by the arcs of great circles and never by small circles. The arcs BC, CA, AB of the spherical triangle ABC are known as the sides of the triangle and are denoted by the a, b, c. Since they are respectively proportional to the angles BOC, COA and AOB that they subtend at O, they are measured by these angles in degrees and minutes of arc. B C ’ O A ’ A C B ’ Figure 6.6 The angles of the spherical triangle ABC are the angles between the planes of the three great circles and are denoted by A, B, C where A is the angle between the planes ABA′B′ and ACA′C′; B is the angle between the planes, ABA′B′ and BCB′C′ and C is the angle between the planes ACA′C′ and BCB′C′. The three sides and the three angles together are called the six elements of the spherical triangle. Polar Triangle Let ABC, A′B′C′ be two spherical triangles on the surface of a sphere with centre at the point O such that A′ is the pole of the great circle BC and lies on the same side of it as the vertex A, B′ is the pole of the great circle AC on the same side of it as the vertex B and C′ is the pole of the great circle AB on the same side of the circle as the vertex C. Then the triangle A′B′C′ is known as the polar triangle of the spherical triangle ABC. The original triangle ABC is called the primitive triangle of the polar triangle. It can be readily seen that if A′B′C′ be the polar triangle of the triangle ABC, then ABC will be the polar triangle of A′B′C′. Now produce B′O to the point R, and since OR is perpendicular to the plane COA, and OA′ is perpendicular to the plane BOC, the angle ROA′ is equal to the angle 130 between the planes COA and BOC and is therefore equal to the angle C. Spherical R Trigonometry O A′ A b ’ b C ’ C ’ C C a ’ a B ’ B Figure 6.7 Let a′, b′, c′ be the sides of the spherical triangle A′B′C′, then cBOAAOR′′′=∠ =π−∠ ′ =π−C (from above) Similarly it can be shown that bBa′′=π−and =π−A As ABC is the polar triangle of A′B′C′, it follows that aAbBc=π−′;; =π−′′ =π−C By transposing we get A′′′=π−aB;; =π− bC =π− c The above relations connecting the sides and angles of a polar triangle with those of the sides and angles of its primitive can be put in the form of the following statements. (a) The sides of the polar triangle A′B′C′ are the supplements of the corresponding angles of its primitive triangle ABC. (b) The angles of the polar triangle A′B′C′ are the supplements of the corresponding sides of its primitive triangle ABC. In a Spherical Triangle ABC (a) a + b + c < 360o (b) A + B + C > 180o, but < 540o Proof (a) Consider the spherical triangle ABC.