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Quadrantal Spherical Triangles.
103. If the triangle DFE (fig. page 80) be the supplement of ABC, which is supposed to have the angle B right, then one side DF will be a quadrant (34); therefore DFE is called a quadrantal triangle.
As the sine, cosine, and tangent of an arc are the same in magnitude as the sine, cosine, and tangent of its supplement, the equations for the triangle ABC will apply to its supplemental triangle DEF. Therefore a quadrantal triangle may be resolved by the preceding rules, if we consider as circular parts, the two angles adjacent to the side of 90°, the comple ment of the other angle, and the complements of the other two sides. Hence, in the triangle ZPS, if the side Z ZS=90°, the circular parts will succeed one another in this order; the angle Z, the comp. ZP, the comp. P, the comp. SP, the angle S. Consequently all the equations for the solution of a quadrantal triangle are contained in the following table.
tan. S x cot. ZP s. P x s. SP Stan. Zx cot. SP s. P x s. ZP tan. Z x cot. P cos. S x s. SP Stan. S x cot. P cos. Zx s. ZP
cot. SP x cot. ZP cos. S x cos. P
104. When any two parts are given to find a third, the case may be resolved by the rules in art. 100, or by seeking that equation in the table, which contains the three parts. If SP and ZP be given to find the angle P, these three parts are found in case 5, and the equation is
Rx cos. Pcot. SP x cot. PZ; therefore
log. cos. P= log. cot. SP+ log. cot. PZ-10.
105. Def. The angle opposite to the quadrantal side is called the hypothenusal angle, and the other parts are called simply the sides and angles.
Affections of a Quadrantal Triangle.
106. The sides are of the same kind as their opposite angles, and conversely.
107. The hypothenusal angle is greater or less than 90°, according as a side and its adjacent angle, or the two sides, or the other two angles, are like or unlike.
108. An angle at the quadrant (or arc of 90°) is obtuse or acute, according as its adjacent side and the hypothenusal angle, or the other angle and the hypothenusal angle, are like or unlike.
109. A side is greater or less than 90°, according as its adjacent angle and the hypothenusal angle, or the other side and the hypothenusal angle, are like or unlike.
Other Properties of Quadrantal Triangles.
110. If the hypothenusal angle be 90°, one of the other angles and its opposite side will be each 90°; and the other side and angle will be measured by the same number of degrees. 111. If an angle, or a side, be 90°, the opposite side, or angle, and the hypothenuse will be each 90°; and the other angle and side will be measured by the same number of degrees.
112. If an angle at the quadrant be less than the hypothenusal angle, their sum will be less than 180°; and if greater, their sum will be greater than 180°.
113. If a side be less than its opposite angle, their sum will be less than 180°; and if greater, their sum will be greater than 180°.
114. The difference of the two sides is less than 90°, and their sum is greater than 90°, and less than 270°.
115. Each of the three angles is equal to or less than 90°; or two angles are greater than 90°, and the third angle is less than 90°.
116. The proofs of these properties follow immediately from the properties of a right-angled spherical triangle (p. 80). If ABC be a spherical triangle, right-angled at B (fig. p. 80), then its supplemental triangle DEF will have one side DF,
opposite to B, equal to 90°: and those parts which are of the same or different affections in the triangle ABC, will have their corresponding parts of the same or different affections in the triangle DEF. By substituting 180°-one side or angle in DEF for the supplemental angle or side in ABC, we shall obtain the limits of the angles and sides in DEF.
Because Rx cos. AC cos. AB x cos. BC (100), if AC be less than 90°, its cosine will be positive; therefore R x cos. AC, or cos. AB x cos. BC, is positive. Hence AB, BC must be both less or both greater than 90°; for in the first case their cosines will be positive, and in the second negative (35 Pl. Tr.); therefore in both cases their product will be positive, as it ought. If AC be greater than 90°, its cosine is negative; therefore cos. AB x cos. BC is negative. Hence one of the sides AB, BC must be less than 90°, and the other greater; for then the cosine of one will be negative, and the cosine of the other positive; therefore their product will be negative, as it ought.
Again, R xs. BC= cot. Cx tan. AB (100). Now s. BC is always positive, therefore cot. C x tan. AB is always positive; consequently AB and C must both be less or both greater than 90°; for in the first case the signs of cot. C and tan. AB will be positive, and in the second negative; therefore in both cases their product will be positive, as it ought. Hence the angles are of the same affections as their opposite sides.
117. Both in right-angled and in quadrantal triangles, if the value of the quantity required be expressed by its sine, it is impossible to determine whether that quantity ought to be less or greater than a quadrant; for the sine of an arc is the same as the sine of its supplement (34 Pl. Tr.). Hence the case is said to be ambiguous. In a right-angled spherical triangle the case is ambiguous when a side and its opposite angle are given to find any of the other parts; and in a quadrantal triangle the case is ambiguous when a side and its opposite angle are given to find any of the other parts, the given side not being the quadrantal side.
In all other cases the ambiguity is taken away by considering whether the sign of the quantity required is positive or negative (110 Pl. Tr.). In the equation R x cos. C: cot. AC x tan. BC, if the angle C and the side AC be given, and
BC be required, then tan. BC =
Rx cos. C
Now if C and
AC be of the same affection, tan. BC is +, and therefore BC is less than 90°; but if C and AC be of different affections, tan. BC is and therefore BC is greater than 90°.
In like manner, in all other cases where the quantity required is not expressed by its sine, the cosine, tangent, and cotangent indicate an arc less or greater than 90°, according as the sign of those quantities is positive or negative.
The ambiguity is often taken away by this property of a right-angled spherical triangle, that the sides are of the same affections as their opposite angles (35).
Solution of the Cases of Right-angled Spherical Triangles.
118. In a right-angled spherical triangle, of the three sides and the three angles any two being given, beside the right angle, which is a constant quantity, the other three may be found.
The different cases, or the varieties, which may happen in the solution of right-angled spherical triangles are sixteen. But these varieties may be reduced to six, if they be restricted to such as depend upon the same principles for their solution. Given, to find the other parts,
1. The hypothenuse and an angle,
6. The two oblique angles.
119. These six cases may be resolved by the different propositions for the resolution of right-angled spherical triangles, or by the three following general theorems, which can be remembered more easily than if they were expressed separately.
1. R x s. either side
2. R x cos. either angle =
3. R x cos. hyp.
s. its op. angle X s. hyp.
tan. its adj. side x cot. hyp.
cos. one side x cos. other side.
cot. one angle X cot. other angle. To apply these equations to the solution of every case of right-angled triangles, they must be converted into analogies. 120. In the following table the first column contains the parts given, the second, the parts required, and the third, the proportions by which they are found.