-2 + · · · + n (n−1) (eio)2 (− e−io)n−2 + neio (— e~ío)n−1 +(-e-io)n. 12 e ̄io) When n is odd, the number of terms in the series is even, and when n is even, the number of terms in the series is odd. Therefore, when n is odd, the terms can be grouped in pairs, the first with the last, the second with the last but one, etc. But, when ʼn is even, there will be a certain number of pairs and one extra term, which is the middle term of the series. From this series, general formulas can be derived for expressing sin "0 as a series of sines or cosines of multiples of 0. By using (5) of Art. 101, cos "e can be dealt with in a similar manner. Here special cases only will be given. However, from these and other special cases, laws can easily be discovered that will determine the coefficients, and multiples of the angles. Example 1. Express sin 0 in sines of multiples of 0. Grouping in pairs, the first with the last, the second with the Example 2. Express sin 0 in cosines of multiples of 0. 1 [ei60-6 ei40+15 ei20 - 20+15 e ̄i20-6 e ̄¡10 +e ̄i Example 3. Express cos3 0 in cosines of multiples of 0. Example 4. Express cos1 0 in cosines of multiples of 0. 1. sin10 2. cos 0 = 3. 128 cos80 4. 64 sin' 0 5. sin Prove the following identities: (cos 404 cos 20 + 3). (cos 70+ 7 cos 50 + 21 cos 3 0 + 35 cos 0). = cos 80 + 8 cos 60+28 cos 40+ 56 cos 20 + 35. = = 35 sin 0 21 sin 30+ 7 sin 50 sin 7 0. 103. Hyperbolic functions. In Art. 43, note, the trigonometric functions were called Circular functions because of their relation to the arc of a circle. There is another set of functions whose properties are very similar to the properties of the trigonometric functions. Because of their relation to the hyperbola, they are called hyperbolic functions. They are defined as follows: (1) Hyperbolic sine x (written sinh x) = (2) Hyperbolic cosine x (written cosh x) et = e-t ex + ex 2 In these formulas e is the base of the Napierian system of logarithms, and so stands for the number 2.7182818 . . . From the definitions, the following relations are evident: 104. Relations between the hyperbolic functions. Squaring (1) and (2) and subtracting the second from the first, sinh 2 cosh y + cosh æ sinh y = } [erts – e-(+)]. sinh x cosh y + cosh x sinh y. 105. Relations between the trigonometric and hyperbolic functions. If in (4) of Art. 101 we substitute i0 for 0, 106. Expression of sinh x and cosh x in a series. Computation. By definition and by (1) of Art. 101, Series (1) and (2) for sinh x and cosh x are convergent for all real values of x. Therefore, for any real value of x the hyperbolic functions of x can be computed. CHAPTER IX SPHERICAL TRIGONOMETRY 107. Spherical trigonometry investigates the relations that exist between the parts of a spherical triangle. For convenience a few of the definitions and theorems of spherical geometry are stated here. 108. The section of the surface of a sphere made by a plane is a great circle if the plane passes through the center of the sphere, and a small circle if the plane does not pass through the center of the sphere. The diameter of a sphere perpendicular to the plane of a circle of the sphere is called the axis of that circle. The points where the axis of a circle of a sphere intersects the surface of the sphere are called the poles of the circle. 109. Spherical triangle. A spherical triangle is the figure on the surface of a sphere bounded by three arcs of great circles. The three arcs are the sides of the triangle, and the angles formed by the arcs at the points where they meet are the angles of the triangle. The angle between two intersecting arcs is measured by the angle between the tangents drawn to the arcs at the point of intersection. If a trihedral angle is placed with its vertex at the center of a sphere, the face planes intersect the surface of the sphere in arcs of great circles which form a spherical triangle. The sides of the spherical triangle measure the face angles of the trihedral angle, and the angles of the triangle are equal to the dihedral angles of the trihedral angle. E A B FIG. 115. In Fig. 115, O is the center of a sphere. O-ABC is a trihedral angle. AB, BC, and CA are arcs of great circles. ABC is a |