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(17.) Trigonometrical, like all other algebraical quantities, are susceptible of different signs under different circumstances. The signs of the sine and cosine are determined by the same rules as the coordinates of a point in analytic geometry. To explain the several changes of sign of the sine and cosine, let C be the centre of a circle whose radius CA is unity, and let CA be the initial position of the revolving radius.

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When the radius coincides with CA, the sine of the arc 0, and the cosine 1. This will be manifest from considering the triangle CPM to be continually changed by the radius CP approaching CA. During this change PM, the sine of the angle continually diminishes, and CM its cosine continually approaches to equality with CA or unity; and when the angle at C actually vanishes, and CP coincides with CA, then PM vanishes, and CM becomes equal to CA.

Hence for all angles terminated by the radius CA, the sine O, and the cosine = 1.

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While the revolving radius moves from CA through the angle ACa, the sine PM continually

at Ca' the cosine vanishes, and the sine coincides with Ca', and ... — — 1.

Through a'CA the sine diminishes continuing negative, and the cosine increases and is positive, thus changing its sign in passing through zero at Ca'.

Thus the several changes which the sine and cosine undergo in one revolution of the radius are evident, and they suffer the same changes every revolution.

By the formulæ found in (10) and (16), it follows that the tangent and cotangent are positive when the sine and cosine have like signs, and negative when they have unlike signs; and by (12) it appears that the sign of the secant is always that of the cosine; and by (13) that the sine of the cosecant is always that of the sine. Thus the signs of the sine and cosine regulate those of all other trigonometrical terms.

SECTION III.

increases, and the cosine decreases until it coin- Of the Relations between Angles and their Sums and cides with Ca, where the sine coincides with Ca, and is. 1, and the cosine vanishes, or = 0.

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Through the angle ACa we shall consider the sine and cosine, both positive, and, in geueral, we shall consider those sines which are measured from the diameter AA' in the direction Ca as positive, and those which are measured in the opposite direction Ca' as negative.

Also, those cosines which are measured in the direction CA we shall consider positive, and those in the opposite direction CA' negative. It will be found that by this arrangement all trigonometrical quantities will change their signs upon passing through zero and infinity. As the radius revolves through the angle aCA' from Ca towards BA', the sine diminishes, but is still positive, and the cosine increases, and is negative. Thus, at Ca the cosine passes through zero, and changes its sign. When the radius coincides with CA', the sine O, and the cosine coinciding with CA' is = — 1. Through the angle A'Ca' the sine increases, and the cosine diminishes, both being negative, the sine changing its sign in passing through 0 at CA'; and at Ca' the cosine vanishes, and the sine coincides with Ca', and '.⋅ — — 1.

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Differences.

(18.) Given the sines and cosines of two arcs to find the sines and cosines of their sum and difference.

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MO'
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sin.'=

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PM
PO'
MO
PO'

PN

sin.(+)=

PO.

cos.(+)=

NO
PO.

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Also, those cosines which are measured in the direction CA we shall consider positive, and those in the opposite direction CA' negative. It will be But PN=DN+ DP. Hence, found that by this arrangement, all trigonometrical quantities will change their signs upon passing through zero and infinity. As the radius revolves through the angle aCA' from Ca towards CA, the sine diminishes, but is still positive, and the cosine increases, and is negative. Thus, at Ca the cosine passes through zero, and changes its sign. When the radius coincides with CA' the sine O, and the cosine coinciding with CA' is — 1. Through the angle A'Ca' the sine increases, and the cosine diminishes, both being negative, the sine changing its sign in passing through 0 at CA'; and

By the similar triangles,

sin.(+)=sin. cos.' + sin.' cos...

Hence,

[1.]

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cos.(a+w')

=

00

cos. cos sin. sin.

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=—cot.(+)cot.(—) [23].

(24.) The sine and cosines of the doublet of an angle may be found by making =<~' in [1] and [2]. This gives,

sin.2=2sin. cos. cos.2, cos. --sin.-2cos.-1. (25.) If in the last formulæ 2 be changed into and therefore into we find the values for the sine and cosine of half an angle.

00

11

sin. (1-cos.) cos. (1+cos.). (26.) The tangent of half an angle may be derived. from these by (10) and (16), and hence we find, sin. 1+cos.

tan.=

1-cos.a sin.

-COS.

=

1 + cos..

SECTION IV.

On the Solution of Plane Triangles.

(27.) In a plane triangle there are six parts, the three sides and the three angles. In general, if the magnitudes of any three of these six quantities be given, the magnitudes of the other three may be computed by the aid of the formulæ and tables of trigonometry.

To guide us in some degree in the determination

Eliminating the sines and cosines by the values of formulæ exhibiting the relations of the parts of for the tangents in (10), we find

tan.(+)=

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sin.+sin.2sin. (+)cos.() sin.sin.2sin. (-)cos.(+) cos."+cos."'=2cos. (+)cos. (a) cos.—cos.—2sin. (@+œ')sin. (-) [17]. (23.) From these four a group of six others may be immediately deduced by division. Let the first Let the first be divided successively by the second, third and fourth; the second by the third and fourth; and the third by the fourth. After this division, the sines and cosines of (~+∞') and }(∞—~) being eliminated by the formulæ found in (10) and (16), we obtain the following formula:

VOL. XVIII. PART I.

a triangle, it is necessary that we should attend to the nature of the tables, by the aid of which these formulæ are to be computed when particular numbers are substituted for their general symbols. These tables are in general logarithmic, and by them, whenever an angle is known, the logarithms of its sine, cosine, tangent, &c. can be determined, and, vice versa, when any of the latter are known, the angle can be found. Formulæ, therefore, in order to be suited to such tables, should be such as are adapted for logarithmic calculation, and therefore their different parts should be united as much as possible by multiplication, division, involution, and evolution; and as little as possible by addition and subtraction.

Further, as it would be impossible in any tables to give the values of the sine, cosine, &c. of angles of all magnitudes, it must frequently happen that the angle, sine, or cosine, &c. which we seek, lies between two successive tabulated angles, sines, or cosines, &c. We can in this case only compute the true value approximately, and the degree of the approximation frequently depends on the formula

I⭑

which we use. A formula which is proper when the angle is great, is often ill suited if the angle to be computed be small. Hence it is necessary in some cases to establish several different formulæ for the solution of the same problem, some being fitted for calculation in the cases where others fail, or give results deviating considerably from the truth.

The Solution of Right-angled Triangles. (28.) Of the six parts of a triangle, it is only necessary to consider four in the case of a rightangled triangle; since the right angle is always given, and one of the acute angles is the compliment of the other.

Let a and b be the sides, c the hypothenuse, and A the angle opposite the side a.

By (8.) we have

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b

=sin. A,

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-=cos.A, := b

=tan.A, =cot.A.;

α

and by plane geometry, a2+b2=c3.

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equation

4abcos.C= (a + b)o — c3, 4absin.C (a — b)2 + c3.

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These equations are sufficient for the solution of By multiplying and dividing these, observing the all cases of right-angled triangles. All questions. as to their solution must come under some of the following four:

1o. Given the two sines, to find the hypothenuse and either angle.

c = √ a2 + b3, •:• lc = \l(a2 + b2)*,

α ...tan. A la- lb.

tan.A ==

b

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we obtain

sin. C 2sin. C cos. C,

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4a basin.oC = [(a+b)—c3] × [—(a—b)o + c3],

tan.3¿C— —(a —b)2 + c2

(a + b)3 — c2

All these results may be easily adapted for logarithmic calculation. Let

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(a + b) — c2 = (a + b + c) (a + b—c), —a—b)2 + c2 = (a + c—b) (c + b − a). Hence the four equations already obtained, when divided by 4ab, give

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cos. 24C =

ab

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which are all suited to logarithmic computation. (30.) The formule which we have now established are sufficient for the solution of oblique plane triangles. The data in all such problems may be reduced to the following:

1o. Given two sides and the angle opposite to one of them.

2o. Given two sides and the angle included by them.

3o. Given two angles and the side opposite to one

of them.

4o. Given two angles and the side between them. 5o. Given the three sides.

We shall consider these problems successively. 1o. Let a and b be the given sides, and A the given angle.

* The radius should be introduced when it is not 1 (as here supposed.)

sin. B

b

α

sin. A, •.• Isin.B= lb + Isin. A— la. By this formula, sin. B becomes known, but, except in certain cases, the angle B will be equivocal. Since an angle and its supplement have the same sine, and neither necessarily exceeding two right angles, they may be each an angle of a triangle; therefore we can only determine that B is either of two angles which are known and supplemental, but it is in general impossible to decide which it is. If ba, the angle B must be acute. It may therefore be found in this case.

If it happen to be known that b < c, though c itself be not known, the same conclusion follows.

The angle B being computed, the angle C becomes known, and thence the side c may be computed by

C =

sin.C sin.A

a.

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The value of A+B is found by subtracting C from 180°. Hence A-B becomes known by the preceding formula, and from this and A+B the values of A and B are derived.

Having found A and B, C may be determined by

sin.c=a

sin.C sin.A

(4.) All great circles bisect each other, and a secondary bisects all parallels to its primary.

1°. The intersection of the planes of two great circles is necessarily a diameter of the sphere, and a common diameter of both circles. Hence they necessarily bisect each other.

2°. By (3.) the plane of a secondary passes through the axis of its primary, and therefore through the centres of all parallels to the primary. Hence it bisects all such parallels.

(5.) The angle under two great circles is equal to the angle under their planes.

The intersection of the planes of two great circles is the common diameter joining the points of intersection of their circumferences. Tangents to the circles drawn from these points of intersection are necessarily in the planes of the circles and perpendicular to their common diameter; hence the angle under these tangents is at the same time the angle under the circles, and the angle under their planes.

(6.) The angle under two great circles is equal to the distance between their poles.

The axes of the great circles being perpendicular to their planes are inclined at the same angle as the But the angle under the axes planes of the circles.

is obviously measured by the arc which joins their extremities, that is, by the distance between the

poles of the great circles.

(7.) The angle under two great circles is measured by the arc of a common secondary intercepted between them. For this arc and the distance between the

3o. Let the angles be A, B, and the given side a. poles have a common complement. Hence

b =

sin. B sin.A

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SPHERICAL TRIGONOMETRY.
SECTION I.

Of Circles on a Spherical Surface.

(1.) By the principles of solid geometry it is proved that if a sphere be intersected by a plane, the section is a circle. If the plane pass through the centre of the sphere, the section is called a great circle; if not, it is a lesser circle.

(2.) That diameter of the sphere, which is perpendicular to the plane of any circle of the sphere, is called the axis of that circle, and the extremities of this diameter are called the poles of the circle.

(3.) A great circle, whose plane is perpendicular to any circle, is said to be secondary to it. It is evident that the planes of all secondaries to a circle pass through its axis, and their circumferences pass through its poles.

SECTION II.

Of Spherical Triangles.

(8.) Def. Three points upon the surface of a sphere being connected by arcs of great circles, the figure formed on the surface by these arcs is called a spherical triangle.

(9.) Any two sides of a spherical triangle taken together are greater than the third side.

For if the radii of the sphere be drawn to the three angles, they will form at the centre a solid angle bounded by three plane angles which are equal to the sides of the proposed triangle. It is proved (Eucl. lib. xi. prop. 20), that any two angles forming such a solid angle must be together greater than the third.

(10.) The sum of the three sides of a spherical triangle is less than the circumference of a great circle.

For let any two of the sides a, b, be produced through the third side c until they meet again. The produced parts a and b will, with the third side c, form a triangle. Hence by the last proposition,

bc,

•.• 2% > a+b+c.

(11.) If the intersections of three great circles be the poles of three others, the intersections of the latter will be the poles of the former.

Let a, b, c, be three great circles, and a', b', c',

three others. The intersections of a and b are the poles of c', the intersection of the planes of a and b is the axis of c' and is perpendicular to every line in the plane of c'. Therefore it is perpendicular to the intersection of the planes of a' and c'.

Also, for the same reason, the intersection of the planes of b and c is perpendicular to the intersection of the planes of c' and a'. Since then the intersection of the planes of c' and a' is perpendicular to the intersection of the planes of b and c, and also to that of the planes b and a, it is perpendicular to two lines in the plane b, and is therefore perpendicular to the plane b itself. Euc. lib. xi. Hence the intersection of the planes a' and c' is the axis of b, and the intersections of the circles and c' are therefore the poles of b.

In like manner it may be proved that the intersections of b' and c' are the poles of a, and those of a' and b' the poles of c.

(12.) If the poles of the sides of a spherical triangle be joined by arcs of great circles, the sides of the triangle so found will be equal to the angles of the former, and vice versa.

Let a, b, c, be the sides of the first, and A, B, C, the angles. By (6.) the sides of the second triangle must be equal to the supplements of the angles of the first, and by (11.) the sides of the first must be equal to the supplements of the angles of the second. Two triangles thus related are called polar triangles.

(14.) Def. Two semicircles being described upon the same diameter of a sphere, that part of the surface of the sphere which they include is called a lune:

(15.) Def. The common diameter of the semicircles is called the axis of the lune.

(16.) Def. The angle under the planes of the semicircles is called the angle of the lune. (17.) Lunes of the same sphere, whose angles are equal, have equal surfaces.

For if their axes be supposed to be placed in coincidence as well as the planes of two of their semicircles, the planes of the other two semicircles being turned in the same direction, must coincide, since the angles of the lunes are equal. Therefore the semicircles which bound the lunes will coincide, and therefore their surfaces will necessarily also coincide and be equal.

(18.) Given the surface of the sphere and the angle of a lune, to determine its area.

Let be the angle of the lune and S the surface of the sphere. Let the angle be divided into any number of parts n. It is plain that the lune may be divided into a number (n) of equal lunes; the angles of which will be Now as often as -- is contained in 360 degrees, so often will one of these lunes be contained in the whole surface of the sphere. Hence it appears that the lune whose angle is, bears to the surface S the ratio : 360°. If L be the lune

n

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n

The surface of a sphere is proved to be equal to

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(21) Cor. 3. The whole surface of the sphere consists of four rectangular lunes.

(22.) Cor. 4. A secondary to the sides of a lune divides it into two equal rectangular isoceles triangles, whose vertical angles are those of the lune. (23.) Given the surface of the sphere, to determine the area of a given triangle.

Let each pair of sides of the triangle be produced through the extremities of the third side until they intersect. There will thus be three lunes formed, whose angles will be the three angles of the triangle, and their surfaces respectively will be 2r A, 2r2B, 2roC,

r being the radius of the sphere, and A, B, C, the angles of the triangle (171.)

In the surfaces of these lunes the given triangle is three times repeated, and with it three of the eight triangles into which the whole sphere is divided by the circles which form the given triangle. The surface of the hemisphere is equal to those three triangles together with the given one. Hence the sum of the three lunes exceed the hemisphere by twice the area of the given triangle. Therefore, if D be the area of the triangle

D=r2(A + B + C)—r3x, ·.·D=ro[(A + B + C)—”.]

Hence the area of a spherical triangle is equal to that of a lune of the same sphere whose angle is equal to half the excess of the sum of the three angles of the triangle above two right angles.

In the formula just obtained, the angles A, B, C,T, are related to the radius unity.

If they should be expressed in seconds, it will be necessary to divide each angle by 206265 (4.)

(24.) Cor. 1. The areas of triangles on the surface of the same sphere are proportional to the excess of the sums of their angles above two right angles.

SECTION III.

Trigonometrical Formulæ expressing relations between the Sides and Angles of a Spherical Triangle.

(25.) The analytical formulæ, which expresses all the various trigonometrical relations of the sides and angles of a spherical triangle, although very numerous, and many of them apparently unconnected, may, nevertheless, be all derived from one formula, which may be considered as the foundation of the whole structure of spherical trigonometry. This formula, therefore, may be regarded in spheri cal trigonometry in the same point of view as that for the sine of the sum of two arcs in plane trigonometry, and, as in that case, we shall establish it

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