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But AD+DB AB, when the perpendicular falls within the triangle; by substituting tangents for cotangents, we have therefore the following theorem of such frequent use, and which may be announced as follows:
84. In any oblique-angled spherical triangle, on the base of which we have let fall a perpendicular arc, and which lies within the triangle, the tangent of the half base is to the tangent of half the sum of the other two sides, as the tangent of the half difference of these sides is to the tangent of half the difference of the segments of the
Observe that if the arc falls without the triangle, we should have AD-BA =AB; and we must then merely substitute the tangent of half the sum of the segments, to the tangent of the half difference of these segments.
85. Returning to the complemental triangle CDE, we have
sin CD:: sin C; sin DE Therefore R cos AC:: sin C; cos B
That is to say, in any right-angled spherical triangle, radius is to the cosine of one of the sides of the right-angle, as the sine of the oblique angle opposite to the other side is to the cosine of the other oblique angle.
86. And consequently if we let fall an arc perpendicularly on the base of an oblique-angled
triangle, the sines of the angles at the summit will be proportional to the cosines of the angles of the base.
87. Let there now be the triangle ABC, right-angled at A, and let there be drawn the tangents BP and BQ: the latter to the hypothenuse
BC, and the first to the side BA. If
intersection BQ will consequently be
perpendicular to the same plane; and the triangle BPQ, rightangled at P, will be similar to the triangle CFE. Thus we shall
FE: GE: BQ : BP
R: cos B: tang BC: tang AC
Therefore in any right-angled spherical triangle, radius is to the cosine of one of the oblique angles, as the tangent of the hypothenuse is to the tangent of the side opposite to the other angle.
88. If the trian
gle ABC is obliqueangled, we may let fall on its base the
cos ACD: cos BCD :: tang BC: tang AC
That is, if we let fall an arc perpendicularly on the base of an oblique-angled spherical triangle, the cosines of the angles at the summit will always be reciprocally proportional to the tangents of the adjacent sides.
89. And as in the complemental triangle CDE, we have R: cos D:: tang CD: tang DE
It is evident that we also have
R: sin AB:: cot AC: cot B:: tang B: tang AC Therefore, in any right-angled triangle, radius is to the sine of one of the sides of the right-angle, as the tangent of the oblique angle opposite to the other side is to the tangent of this last side.
90. The same triangle CDE gives R: sin CE:: tang C: tang DE therefore, R: cos BC;: tang C: cot B
Whence it follows that in every right-angled spherical triangle, radius is to the cosine of the hypothenuse, as the tangent of one of the oblique angles is to the colangent of the other angle.
APPLICATION OF THE PRECEDING PRINCIPLES AND PROPORTIONS.
91. By means of the propositions which we have just demonstrated, we can very easily solve any spherical triangle, provided three of its parts are given. We shall commence with right-angled triangles.
And first, as the right-angle is always given, it is sufficient to know two of the other five parts, that make up these triangles. Again the number of combinations of m quantities, taken two m. (m—1) (Page 111). It is eviby two, is expressed generally by 2
dent, therefore, 1o, that in this point of view, right-angled spherical triangles present ten of these combinations. But, as for each combination we have three quantities to determine, it is clear 2dly, that all the possible varieties in the solution of right-angled spherical triangles are 30 in number.
The following Table contains them all, the angle A is supposed to be the right-angle; and the other two angles are denoted indifferently by B or C.
92. The construction of this Table is wholly founded upon two propositions already demonstrated.
We shall place them once more F
under the eye of the student.
PROP. 1. In every right-angled spherical triangle, radius is to the sine of the hypothenuse, as the sine of one of the oblique angles is to the sine of the side opposite to it (81).
PROP. 2. In every right-angled spherical triangle, radius is to the sine of one of the sides of the right-angle, as the tangent of the oblique angle opposite to the other side is to the tangent of that side (89).
Sometimes these proportions are applied immediately to the triangle ABC, and sometimes we must have recourse to one of the complemental triangles CDE, BFG, in order afterwards to transport the results to the triangle ABC, as will be shewn in several examples.
For the solution of all the possible cases of a spherical triangle ABC, right-angled at A.
The part required is less than 90°.
R: sin BC::sin B sin AC If B 90°
R: sin BC::sin C: sin AB
If BC and B are of the same species
If BC and C are of the same species
R sin AC::tang C: tang AB│If C 90°
R: cos AC::sin C: cos B
tang B: tang AC::R: sin AB
If AC and C are of the same species
If BC and AC are of the same species
R: sin AB::tang B: tang AC
R: cos AB::cos AC: cos BC
AC cos AB cos BC::R: cos AC
If BC and AB are of the same species
If B and C are of the same species
To facilitate the application of this Table, let the hypothenuse BC-81° Is, and the angle B-37° 19′; required the side AC, opposite to the angle B. To solve this case, we must employ the first proportion in the Table, and say R sin BC:: sin B: sin AC
Therefore AC-36° 48', or 143° 12', which is its supplement. To decide which of these values is the right, we must call to mind that the side AC must be of the same species as the angle B oppoto it (74.) By way of reminding the calculator of the conditions upon which depend the results which he seeks for, they are inserted in the last column of the Table.
93. Supposing still the same hypothenuse BC and the same angle B to be given, and that the adjacent side AB were required; it would be easy in the first place to find the side AC, as before, and afterwards to employ the proportion
tang B: tang AC :: R; sin AB.
But this process introduces two proportions into a computation which may be managed by only one. For in the complemental triangle CDE we have
R: sin DE: : tang D: tang CE,
and therefore transporting this proportion to the triangle ABC, it will become
or perhaps better,
R: cos B: cot AB : cot BC,
R: cos B: tang BC: tang AB,
log radius taken negatively
log cos B (37° 19′)
log tan BC (81° 13')
log tang AB
This last logarithm corresponds to 79° or 101°; but as BC and B are of the same kind, we must take the first value: Consequently the side AB is 79°, or calculating as far as seconds, 79° 0 20".
N. B. When the logarithm of radius is to be subtracted, we shall simply write it 10; and in the aggregation of the three terms which form the proportion, the unit surmounted with the negative sign must be considered as negative and subtracted.
To find the angle C, we must have recourse to the triangle CDE, in which we have
sin CER: tang DE: tang C,
whence we obtain by substitution,
cos BC: R:: cot B: tang C
By logarithms, we obtain the angle C as follows:
neg.log of radius
9.183834 9.882101 =9.065935
log cos BC (81
which gives C-83° 22', and not 96° 38', since BC and B are of the same kind.
94. If instead of the hypothenuse being given, we had known the adjacent side AB, with the same angle B, and had desired the