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and the sign in the first quadrant EB, reckoned in the direction EB, perpendicular to the horizontal diameter AD, being considered to be positive or +, the sign NM in the second quadrant is also +, and those in the third and fourth quadrant NL, EI, are negative or —. So the cosine GB in the first quadrant being positive or +, those in the second and third quadrants perpendicular to the vertical diameter CK, and on the opposite side of it, are negative or those in the fourth quadrant are +.
So AF, the tangent of the arc AB, being +, that of ABDL, being also AF, is +; and that of the arc ACM, or AMI, would lie below A in the opposite direction to AF, and would be negative.
The signs of the cotangents are determined in a similar manner. The secants are the reciprocals of the cosines when radius 1; and hence they have the same signs as the cosines. For a similar reason, the cosecants have the same signs as the sines, and the cotangents the same as the tangents.
Let 7 represent a semicircle, or an arc of 180°, then the signs of these trigonometrical quantities are as in the subjoined table.
| + | +
Tangent. Secant. Cosine. Cotan.
327. It appears from this table that the sines and cosecants of arcs between 0° and 180° are all positive. Hence, when the sine or cosecant of an arc less than 180° is given to find the arc, it is uncertain whether the arc ought to be less or greater than 90°; for if a denote an arc whose sine is
given, and if a 180°, the given sine is also the sine of (180°-a). If, however, the cosine, tangent, cotangent, or cosecant of an arc is given, its species is easily found; for if the cosine, for example, is positive, the arc must be 90°; and if the cosine is negative, the arc must be 90°.
So if an arc a is 180°, and if sina is given, there can be no ambiguity as to its species, for a must be
328. The sine, tangent, and cotangent of a negative arc less than are negative, but its cosine is positive.
The species of the sides and angles of right-angled spherical triangles can also be easily determined, when possible, by means of the following theorems :—
329. THEOREM I.-In a right-angled spherical triangle the sides are of the same affection with the opposite angles; that is, if the sides be greater or less than quadrants, the opposite angles will be greater or less than right angles.
Let ABC be a spherical triangle right-angled at A, any side AB will be of the same affection with the opposite angle ACB.
Case 1.-Let AB be less than a quadrant. Let AE be a quadrant, and EC an arc of a great circle passing through E, C. Since A is a right angle, and AE a quadrant, E is the pole of the great A circle AC, and ECA is a right angle; but ECA is greater than BCA, therefore BCA is less than a right angle.
Case 2.-Let AB be greater than a quadrant; make AE equal to a quadrant, and let a great circle pass through C, E. ECA is a right angle as before, and BCA is greater than ECA, that is, greater than a right angle.
330. THEOREM II.-If the two sides of a right-angled spherical triangle be of the same affection, the hypotenuse
will be less than a quadrant; and if they be of different affection, the hypotenuse will be greater than a quadrant.
Let ABC (last two figures) be right-angled spherical triangles; if the two sides AB, AC, be of the same or of different affection, the hypotenuse BC will be accordingly less or greater than a quadrant.
Case 1.-Let AB, AC, be each less than a quadrant. Let AE, AG, be quadrants; G will be the pole of AB, and E the pole of AC, and EC a quadrant; but CE is greater than CB, since CB is farther off from CGD than CE. In the same manner, it is shown that CB, in the triangle CBD, where the two sides CD, BD, are each greater than a quadrant, is less than CE, that is, less than a quadrant.
Case 2.-Let AC be less, and AB greater than a quadrant (second fig. in 329); then the hypotenuse BC will be greater than a quadrant. For, let AE be a quadrant, then E is the pole of AC, and EC will be a quadrant. But CB is greater than CE, since AGB passes through the pole of BD; that is, BC is greater than a quadrant.
COR. 1.-Hence, conversely, if the hypotenuse of a rightangled triangle be greater or less than a quadrant, the sides will be accordingly of different or the same affection.
COR. 2. Since the angles of a right-angled spherical triangle have the same affection with the opposite sides, therefore, according as the hypotenuse is greater or less than a quadrant, the angles will be of different or of the same affection.
331. The sides are of the same species with the opposite angles.
332. If two sides are of the same species, the hypotenuse is less than a quadrant; and if of different species, the hypotenuse is greater than a quadrant; and, conversely,
333. According as the hypotenuse is greater or less than a quadrant, the sides are of different or of the same species; and also the angles are of different or of the same species."
RIGHT-ANGLED SPHERICAL TRIGONOMETRY.
334. All the cases of this branch of spherical trigonometry can be easily solved by means of the two rules of the
circular parts (invented by Lord Napier), which are as follows:
'If the right angle of a right-angled triangle is omitted, and if the hypotenuse, the two angles, and the complements of the two sides, are called the circular parts; also if any of these parts is called the middle part; the parts contiguous to it, the adjacent parts; and the remaining parts, the opposite parts; then
335. The rectangle under the radius and the cosine of the middle part is equal to the rectangle under the cotangents of the adjacent parts, or to that under the sines of the opposite parts."
That is, if M denote the middle part, A and a the adjacent parts, and O and o the opposite parts, Rad cos Mcot A cot a
Rad cos M = sin O⚫ sin o
When these quantities are supposed to be expressed by numbers, the term product ought to be used instead of rectangle in the enunciation.
These two equations can be converted into proportions by the principle that the product of the extreme terms (the first and fourth) of a proportion is equal to the product of the means" (the second and third). Hence in , if cos M is made the fourth term, radius must be made the first term of a proportion; and cot A and cot a, the second and third indifferently. And, generally,
'If the product of two numbers is equal to that of other two, a proportion can be formed by making the factors of one product the extremes, and those of the other the means; that is, a factor of one of the products is to either factor of the other, as the remaining factor of this product to the remaining factor of the first.
Hence the factor which is required must be the fourth term, and the factor of the same product must be the first term of the proportion."
The various cases of this branch of spherical trigonometry can also be easily solved, by means of the analogies in the table at the end of the spherical trigonometry in the second volume of Geometry of the Educational Course.
The parts of a triangle concerned in a calculation are the
given parts employed and the required part. When of the parts concerned, one of them is contiguous to the other two, make it the middle part, and the other two will be the adjacent parts; and when one of the parts concerned is separated from the other two by intermediate unconcerned parts, make it the middle part, and the other two concerned parts will be the opposite parts.
Besides the rules in articles 329, 330, for determining the species of the parts of a triangle, they may also be ascertained by the following rule :—
336. In any proportion, if the fourth term is a cosine, tangent, or cotangent, and of the arc whose cosines, tangents, or cotangents enter in the first three terms, if one or three are greater than a quadrant, so is the fourth term ;" or,
'If the signs of one or of three of the terms are negative, the sign of the fourth is also negative, and this part is greater than a quadrant; in other cases its sign is positive, and this part is less than a quadrant."
This rule is evident from the algebraical signs of the terms of a proportion, and the rules for algebraical multiplication and division.
337. Case I.-Given the hypotenuse, and one of the angles, to find the other parts.
EXAMPLE. In the spherical triangle ABC, right-angled at B, the hypotenuse AC is 64°, and the angle C 46°; what are the remaining parts ?*
To find BC.
When angle C is the middle part (335), AC and the complement of BC are the adjacent parts, and Rcos M = cot A cot a becomes R cos Ccot AC cot of comp. of BC.
Or R⚫ cos C = cot AC tan BC; and as BC is wanted, the proportion must be (335)
Cot AC: R= cos C: tan BC.
*The reader is referred to the figures under the preceding head (see fig. art. 304.)