Spherical Trigonometry
SPHERICAL TRIGONOMETRY
DEFINITION OF TERMS The sphere is the set of all points in a three-dimensional space such that the distance of each from a fixed point is constant. The fixed point and the given distance are called the center and the radius of the sphere respectively. The intersection of a plane with a sphere is a circle. If the plane passes through the center of the sphere, the intersection is a great circle; otherwise, the intersection is a small circle. A line perpendicular to the plane of a circle and through the center of the sphere is called the axis of the circle. The intersection of this axis and the sphere are called the poles of the circle. Opposite ends of a diameter are identified as antipodal points. Two great circles intersecting in a pair of antipodal points divide the sphere into four regions called lunes. Thus a lune is bounded by the arcs of two great circles. The polar distance (in angular units) of a circle is the least distance of a point on the circle to its pole. Two distinct points on the sphere which are not ends of a diameter divide the great circle into two arcs. The shorter arc is called the minor arc.
SPHERICAL TRIANGLES A spherical triangle is that part of the surface of a sphere bounded by three arcs of great circles. The bounding arcs are called the sides of the spherical triangle and the intersections of these arcs are called the vertices of the spherical triangle. The angle formed by two intersecting arcs is called a spherical angle. Like a plane triangle, the spherical triangle has also six parts – three angles and three sides. The sides a, b, and c are measured by the corresponding faces of the trihedral angle.
Important Propositions from Solid Geometry: 1. If two sides are equal, the angles opposite are equal and conversely. 2. If two sides are unequal, the angles opposite are unequal and the greater side is opposite the greater angle and conversely. 3. The sum of any two sides is
DEFINITION OF TERMS The sphere is the set of all points in a three-dimensional space such that the distance of each from a fixed point is constant. The fixed point and the given distance are called the center and the radius of the sphere respectively. The intersection of a plane with a sphere is a circle. If the plane passes through the center of the sphere, the intersection is a great circle; otherwise, the intersection is a small circle. A line perpendicular to the plane of a circle and through the center of the sphere is called the axis of the circle. The intersection of this axis and the sphere are called the poles of the circle. Opposite ends of a diameter are identified as antipodal points. Two great circles intersecting in a pair of antipodal points divide the sphere into four regions called lunes. Thus a lune is bounded by the arcs of two great circles. The polar distance (in angular units) of a circle is the least distance of a point on the circle to its pole. Two distinct points on the sphere which are not ends of a diameter divide the great circle into two arcs. The shorter arc is called the minor arc.
SPHERICAL TRIANGLES A spherical triangle is that part of the surface of a sphere bounded by three arcs of great circles. The bounding arcs are called the sides of the spherical triangle and the intersections of these arcs are called the vertices of the spherical triangle. The angle formed by two intersecting arcs is called a spherical angle. Like a plane triangle, the spherical triangle has also six parts – three angles and three sides. The sides a, b, and c are measured by the corresponding faces of the trihedral angle.
Important Propositions from Solid Geometry: 1. If two sides are equal, the angles opposite are equal and conversely. 2. If two sides are unequal, the angles opposite are unequal and the greater side is opposite the greater angle and conversely. 3. The sum of any two sides is