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PART TWO

SPHERICAL TRIGONOMETRY

CHAPTER V

RIGHT SPHERICAL TRIANGLES

DEFINITIONS AND ELEMENTARY PRINCIPLES

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118. Spherical Triangle. Let A, B, C, be any three points on the surface of a sphere whose centre is O. Through (1) 0, A, B, (2) 0, A, C, (3) 0, B, C, pass planes forming the triangular spherical pyramid whose vertex is 0, and whose base is ABC, then the figure ABC is a spherical triangle, of which the three sides, AB, AC, BC, are arcs of great circles, and the three angles, O A, B, C, are equal to the plane angles which measure the diedral angles of the pyramid.

A

FIG. 55.

α

C

119. Spherical Trigonometry treats of the trigonometric relations among the sides and angles of a spherical triangle. These relations are not affected by varying the radius of the sphere; that is, they are independent of the radius. Hence, for simplicity, the radius, in this book, will always be considered unity or 1. Under this supposition the relative lengths of the sides AB, BC, AC, are the circular measures of the face angles of the pyramid AOB, BOC, AOC, respectively.

120. Notation. As in Plane Trigonometry, the angles of triangles will be denoted by A, B, C, and the opposite sides by a, b, c. The sides are usually expressed in degree measure; that is, a, b, c, stand for the numbers of degrees in the arcs CB, AC, AB, or the angles COB, AOC, AOB.

121. Parts. Every spherical triangle has six parts, - three sides and three angles. Any two parts are said to be of the same or different species according (1) as they are both less or greater than 90°, or (2) as the one is less and the other greater than 90°.

122. A spherical triangle is called trirectangular when it has three right angles, and birectangular when it has two right angles.

123. Geometric Properties. This book treats only of spherical triangles whose parts are each less than 180°. Of these the following properties are proved in Geometry :

1. The sum of any two sides of a spherical triangle is greater than the third side.

2. In any spherical triangle, the greater side lies opposite the greater angle; and, conversely, the greater angle lies opposite the greater side.

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3. The sum of the sides of a spherical triangle is less than 360°.

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4. The sum of the angles of a spherical triangle is greater than 180°, and less than 540°.

5. If A'B'C' is the polar triangle of ABC, that is, if A, B, and C are the poles of the sides B'C', C'A', and A'B', respectively, then, conversely, ABC is the polar triangle of A'B'C'.

6. In two polar triangles, each angle of one is measured by the supplement of the side lying opposite the homologous angle of the other; that is,

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124. Let C be the right angle of the right spherical triangle

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Since BE and DE are 1 OA, ▲ DEB = ▲ A;

hence

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cos c sin b cos A = cos a (cos b tan b) (sin c cot c) sin B

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125. Generality of the Formulas. The preceding ten formulas have been proved only for the case when the legs (a, b) are both <90°. It therefore remains to be shown that they are also true (1) when one of the legs is < 90° and the other > 90°, and (2) when both legs are > 90°.

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In Fig. 58, let a > 90° and b < 90°, and in Fig. 59, let a > 90° and b> 90°. In both figures, precisely as in Fig. 57, draw BD 1 OC (produced) and BE 1 OA (produced, if necessary), then draw ED. Just as in Art. 124, ED will be BD and OA, and therefore OE = cos c, DEB = 2 A.

BD = sin a, OD = cos a, BE = sin c,

Consequently the proofs under Art. 124 apply, word for word, to

126. Relations with Respect to Species. I. It is evident from [73] that cos c is positive when cos a and cos b have the same sign, and negative when they have opposite signs; therefore the hypotenuse (c) of a right triangle is acute or < 90° when the two legs (a, b) are of the same species, and obtuse or > 90° when they are of different species. II. Since each part is <180°, sin a is always positive. It is evident then from [79] that cot B or tan B and tan b always have the same sign; that is, either oblique angle and its opposite side are always of the same species.

127. Napier's Circular Parts and Rules. Formulas [73] to [82] are used in solving right spherical triangles, and the difficulty of remembering and producing the right formula for any given case is largely removed by the use of Napier's Circular Parts and Rules, which we will now explain.

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128. Circular Parts. way, as in Fig. 60, we hypotenuse and the

b

FIG. 61.

Instead of writing the triangle in the usual write it as in Fig. 61; that is, instead of the two oblique angles their complements are employed, the right angle C being omitted. The five parts, a, b, B, are Napier's Circular Parts, and with re

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A, 90°

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spect to any part, as 90° A, the parts standing next to it in the figure, 90° c and b, are called the adjacent parts, and the other two parts, 90°-B and a, the opposite parts. With respect to the part 90°c, 90°-A and 90°- B are adjacent parts, and a and b opposite parts.

129. Rules. I. The sine of any part is equal to the product of the tangents of the adjacent parts.

II. The sine of any part is equal to the product of the cosines of the opposite parts.

NOTE.-These Rules may be more easily remembered by observing that in Rule I the descriptive terms adjacent and tangents contain the vowel a, and in Rule II the descriptive terms opposite and cosines contain the vowel o.

In illustration of these Rules let us, by them, produce formulas [73] to [82]. We may select any part to begin with, as a (Fig. 61), then by Rule

I. sin a tan (90° – B) tan b,

or sin a cot B tan b,

II. sin a = cos (90° — A) cos (90°— c), or sin a = sin A sin c,

which are formulas [79] and [74].

Again, beginning with 90°— B, we have by Rule

I. sin (90°— B) = tan a tan (90° — c), or cos B = tan a cot c, II. sin (90° – B) = cos b cos (90° — A), or cos B: = cos b sin A, which are formulas [77] and [81].

Let the student in a similar manner produce the other six formulas of Art. 124.

That the formulas thus obtained by the Rules are the same as those obtained in Art. 124, is a proof of the correctness of the Rules.

SOLUTION OF RIGHT SPHERICAL TRIANGLES

130. Solvable Cases. It is possible to solve, by means of formulas [73] to [82], any right spherical triangle when two of its parts, not including the right angle, are given.

131. Method without Napier's Rules. Teachers of Trigonometry differ as to the advisability of using Napier's Rules. Some think it preferable to remember the formulas by their analogy to the corresponding ones for right plane triangles. This analogy is very striking when presented as follows:

Denote the corresponding sides and angles of a plane right triangle also by a, b, c, A, B, C, and we have

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