133. Case IV. Given two angles and the included side. This case like the preceding is to be solved by Napier's analogies, using the four forms in a similar manner.

134. Case V. Given two sides and the angle opposite one of them. In this case the angle opposite the other side can be found by the sine law, when the other side and angle can be found by Napier's analogies. For example, given a, b, and a, to find c, B, and y, use the following formulas:

A check is obtained by the agreement in the values of c and y from the different formulas.

Since ẞ is determined from the sin ß, there will be two values of ẞ less than 180°, both of which may enter into a triangle. By the first rule for species, Art. 128, if | 90° - b | > | 90° — a ¦, b and must be the same species. This definitely determines ẞ. Otherwise, both values of ẞ may be admissible. The application of the second rule for species will show whether or not two triangles are possible.

Example. Given a = 148° 34′ 24′′, b = 142° 11′ 36′′, a = 153° 17' 36"; find c, B, and y.

A c1 B1
βι

Ca

a

a

2

FIG. 130.

Here, since | 90° - 142° 11′ 36′′ | < | 90° 148° 34′ 24′′ |, both

values of ẞ may be admissible.

Since (a + b) = 145° 23′ is in (a + B1) 90° 35′ 36′′ and

the second quadrant, as also are

=

1⁄2 (a + B2) = 150° 42', they are of the same species by the second rule for species. Hence both B1 and B2 are admissible values to

use.

The student can complete the solution and find the following values:

This case is the ambiguous case in oblique spherical triangles, and is analogous to the ambiguous case in plane trigonometry. In practical applications, some facts about the general shape of the triangle may be known which will determine the values to be chosen without having recourse to the rules for species. A complete discussion of the ambiguous case may be found in Todhunter and Leathem's Spherical Trigonometry, pp. 80-85.

135. Case VI.

of them.

=

=

=

133°, a = 146°. 103° 18′ 47′′, a =

62° 24′ 25′′, Y1 =

53° 42′ 38′′;

155° 43′ 11′′,

117° 35′ 35′′, Y1⁄2 = 59° 6′ 50′′.

Given two angles and the side opposite one

- In this case use the same formulas as in case V, and apply the rules for species when there is any question as to the number of solutions.

Let r be the radius of

136. Area of a spherical triangle. the sphere on which the triangle is situated, A the area of the triangle, and E the spherical excess.

By spherical geometry, the areas of any two spherical triangles are to each other as their spherical excesses. Now the area of a tri-rectangular triangle is r2, and its spherical excess is 90°.

When all the angles of the triangle are known, the spherical excess and therefore the area are easily computed. If the angles are not all given, but enough data are known for the solution of the triangle, the angles may be found by Napier's analogies, and then the area may be computed by the above formula.

137. L'Huilier's Formula. This is a formula for determining the spherical excess directly in terms of the sides. It may be derived as follows: Since E = a + B+ y 180°,

− 180°) cos (a + B −y + 180°)

180°) cos(a + By + 180°)

sin(a+B) cosy
cos (a + B) + sin y'

by [29] and [31].

cos y cos (a - b)

cos c

siny cos(a+b).

By making these substitutions,

cos c

γ

cot Y

sin (sa) sin (s

c)

(s − a) tan } (s — b) tan } (s — c).

EXERCISES

1. On a sphere of radius 6 in., a =

y = 77° 45′ 32". 2. Given a =

87° 20′ 45′′, B 32° 40′ 56′′, and Find the area of the triangle. Ans. 11.176 sq. in.

=

56° 37',

=

b 108° 14',

с =

75° 29'; find E.

9. Find the area of a triangle having sides of 1o each on the surface of the earth.

Ans. 2070 sq. mi.

APPLICATIONS OF SPHERICAL TRIGONOMETRY

138. Definitions and notations. In all the applications of spherical trigonometry to the measurements of arcs of great circles on the surface of the earth, and to problems of astronomy, the earth will be treated as a sphere of radius 3956 miles.

A meridian is a great circle of the earth drawn through the poles N and S. The meridian NGS passing through Greenwich, England, is called the principal meridian.

The longitude of any point P on the earth's surface is the angle between the principal meridian NGS and the meridian NPS through P. It is measured by the great circle arc, CA, of the equator between the points where the meridians cut the equator.

If a point on the surface of the earth is west of the principal meridian, its longitude is positive. If east, it is negative. A point 70° west of the principal meridian is usually designated as