π sin 30 cos 40 = cosec ( + =) [ tau(n + 1) (0 + =) - tan(+1)] 92. sin a sin 3 a + sin sin+sin sin α 3 α + 2 22 22 PART II. SPHERICAL TRIGONOMETRY. CHAPTER X. FORMULE RELATIVE TO SPHERICAL TRIANGLES. 182. Spherical Trigonometry has for its object the solution of spherical triangles. A spherical triangle is the figure formed by joining any three points on the surface of a sphere by arcs of great circles. The three points are called the vertices of the triangle; the three arcs are called the sides of the triangle. Any two points on the surface of a sphere can be joined by two distinct arcs, which together make up a great circle passing through the points. Hence, when the points are not diametrically opposite, these arcs are unequal, one of them being less, the other greater, than 180°. It is not necessary to consider triangles in which a side is greater than 180°, since we may always replace such a side by the remaining arc of the great circle to which it belongs. 183. Geometric Principles. It is shown in geometry (Art. 702), that if the vertex of a triedral angle is made the centre of a sphere, then the planes which form the triedral angle will cut the surface of the sphere in three arcs of great circles, forming a spherical triangle. Thus, let O be the vertex of a triedral angle, and AOB, BOC, COA its face-angles. We may construct a sphere with its centre at O, and with any radius OA. Let AB, 267 BC, CA be the arcs of great circles in which the planes of the face-angles AOB, BOC, COA cut the surface of this sphere; then ABC is a spherical triangle, and the arcs AB, BC, CA are its sides. Now it is shown in geometry that the three face-angles AOB, B A C BOC, COA are measured by the sides AB, BC, CA, respectively, of the spherical triangle, and that the diedral angles OA, OB, OC are equal to the angles A, B, C, respectively, of the spherical triangle ABC, and also that a diedral angle is measured by its plane angle. There is then a correspondence between the triedral angle O-ABC and the spherical triangle ABC: the six parts of the triedral angle are represented by the corresponding six parts of the spherical triangle, and all the relations among the parts of the former are the same as the relations among the corresponding parts of the latter. 184. Fundamental Definitions and Properties. — The following definitions and properties are from Geometry, Book VIII. : In every spherical triangle Each side is less than the sum of the other two. The sum of the three sides lies between 0° and 360°. The sum of the three angles lies between 180° and 540°. Each angle is greater than the difference between 180° and the sum of the other two. If two sides are equal, the angles opposite them are equal; and conversely. If two sides are unequal, the greater side lies opposite the greater angle; and conversely. The perpendicular from the vertex to the base of an isosceles triangle bisects both the vertical angle and the base. The axis of a circle is the diameter of the sphere perpendicular to the plane of the circle. The poles of a circle are the two points in which its axis meets the surface of the sphere. One spherical triangle is called the polar triangle of a second spherical triangle when the sides of the first triangle. have their poles at the vertices of the second. If the first of two spherical triangles is the polar triangle of the second, then the second is the polar triangle of the first. Two such triangles are said to be polar with respect to each other. Thus : If A'B'C' is the polar triangle of ABC, then ABC is the polar triangle of A'B'C'. In two polar triangles, each angle of one is measured by the supplement of the corresponding side of the other. Thus: B a a' B = 180° - b', C = 180° - c', A = 180° — a', a = 180° — A', This result is of great importance; for if any general equation be established between the sides and angles of a spherical triangle, it holds of course for the polar triangle also. Hence, by means of the above formulæ any theorem of a spherical triangle may be at once transformed into another theorem by substituting for each side and angle respectively the supplements of its opposite angle and side. If a spherical triangle has one right angle, it is called a right triangle; if it has two right angles, it is called a birectangular triangle; and if it has three right angles, it is called a tri-rectangular triangle. If it has one side equal to a quadrant, it is called a quadrantal triangle; and if it has two sides equal to a quadrant, it is called a bi-quadrantal triangle. |