1049. PROB. II. To compute the tangents and secants.

Having, by the foregoing problem, found the sines and cosines, the tangents and secar are easily found from the principles of similar triangles. In the arc AB (fig. 395.), whe BF is the sine, CF or BK the cosine, AH the tangent, CH the secant, DL the cotanger and CL the cosecant, the radius being CA or CB or CD; the three similar triangles CF CAH, CDL, give the following proportions:

I. CF: FB::CA: AH, by which we find that the tangent is a fourth proportion to the cosine, sine, and radius.

II. CF CB:: CA: CH, by which we find that the secant is a third proportional the cosine and radius.

III. BF: FC::CD: DL, by which we find that the cotangent is a fourth proportion to the sine, cosine, and radius.

IV. BF: BC::CD: CL, by which we find that the cosecant is a third proportion to the sine and radius.

Observation 1. There are therefore three methods of resolving triangles, or the cases trigonometry; viz. geometrical construction, arithmetical computation, and instrument. operation. The method of carrying out the first and the last does not need explanation the method is obvious. The second method, from its superior accuracy in practice, is the whereof we propose to treat in this place.

Observation 2. Every triangle has six parts, viz. three sides and three angles. And in a cases of trigonometry, three parts must be given to find the other three. And of the thre

parts so given, one at least must be a side; because, with the same angles, the sides may greater or less in any proportion.

viz.

Observation 3. All the cases in trigonometry are comprised in three varieties only

1st. When a side and its opposite angle are given. 2d. When two sides and the cor tained angle are given. 3d. When the three sides are given.

More than these three varieties there cannot possibly be; and for each of them we sha give a separate theorem.

1050. THEOREM I.

Then the sides of the

When a side and its opposite angle are two of the given parts. triangle have the same proportion to each other as the sines That is,

their opposite angles have.

For let ABC (fig. 397.) be the proposed triangle, having AB the greatest side, and B{ the least.

Take AD as a radius equal to BC, and let

fall the perpendiculars DE, CF, which will evidently be the sines of the angles A and B, to the radius AD or BC. Now the triangles ADE, ACF are equiangular; they therefore have their like sides proportional, namely, AC: CF:: AD or BC: DE, that is, the sine AC is to the sine of its opposite angle B as the side BC is to the sine of its opposite angle A.

c

E

F

B

Fig. 397.

Note 1. In practice, when an angle is sought, the proportion is to be begun with a side opposite a given angle; and to find a side, we must begin with the angle opposite the given side.

Note 2. By the above rule, an angle, when found, is ambiguous; that is, it is not certai whether it be acute or obtuse, unless it come out a right angle, or its magnitude be such a to remove the ambiguity; inasmuch as the sine answers to two angles, which are supple ments to each other; and hence the geometrical construction forms two triangles with the same parts, as in an example which will follow: and if there be no restriction or limitation included in the question, either result may be adopted. The degrees in a table answer ing to the sine is the acute angle; but if the angle be obtuse, the degrees must be sub tracted from 180 degrees, and the remainder will be the obtuse angle. When a given angle is obtuse, or is one of 90 degrees, no ambiguity can occur, because neither of the other angles can then be obtuse, and the geometrical construction will only form one triangle.

Example 1. In the plane triangle ABC,

Let AB be 345 feet,

BC 232 feet,

LA 37 20':

Required the other parts.

First, to the angles at C and B (fig. 398.)

Fig. 398

C

It is to be observed here that the second and third logarithms are added (that is, the sumbers are multiplied), and from the sum the first logarithm is subtracted (that is, division by the first number), which leaves the remainder 9.955127, which, by the table of sines, is found to be that of the angle 115° 36', or 64° 24'.

Herein two angles are given, whose sum is 81° 57'. Therefore 180° — 81° 57′ = L C.

As sine L C

= 980 3'

= 365

Is to AB

So sine / B

2-5622929

9.6218612

=2.1884548

1051. THEOREM II. When two sides and their contained angle are given.

The given angle is first to be subtracted from 180° or two right angles, and the remainder will be the sum of the other two angles. Divide this remainder by 2, which will give the half sum of the said unknown angles; and using the following ratio, we have →

As the sum of the two given sides

Is to their difference,

So is the tangent of half the sum of their opposite angles

To the tangent of half the difference of the same angles.

Now the half sum of any two quantities increased by their half difference gives the greater, and diminished by it gives the less. If, therefore, the half difference of the angles above found be added to their half sum, it will give the greater angle, and subtracting it will leave the lesser angle. All the angles thus become known, and the unknown side is then found by the former theorem.

A

Fig. 339.

D

GF B

For let ABC (fig. 399.) be the proposed triangle, having the two given sides AC, BC, including the given angle C. With the centre C and radius E CA, the less of these two sides, describe a semicircle, meeting the other side of BC produced in E, and the unknown side AB in G. Join AE, CG, and draw DF parallel to AE. Now BE is the sum of the given sides AC, CB, or of EC, CB; and BD is the difference of these given sides. The external angle ACE is equal to the sum of the two internal or given angles CAB, CBA; but the angle ADE at the circumference is equal to half the angle ACE at the centre; wherefore the same angle ADE is equal to half the sum of the given angles CAB, CBA. Also the external angle AGC of the triangle BGC is equal to the sum of the two internal angles GCB, GBC, or the angle GCB is equal to the difference of the two angles AGC, GBC; but the angle CAB is equal to the said angle AGC, these being opposite to the equal sides AC, CG; and the angle DAB at the circumference is equal to half the angle DCG at the centre. Therefore the angle DAB is equal to half the difference of the two given angles CAB, CBA, of which it has been shown that ADE or CDA is the half sum

Now the angle DAE in a semicircle, is a right angle, or AE is perpendicular to AL and DF, parallel to AE, is also perpendicular to AD: therefore AE is the tangent CDA the half sum; and DF, the tangent of DAB, the half difference of the angles to th same radius AD, by the definition of a tangent. But the tangents AE, DF being paralle it will be as BE:BD::AE: DF; that is, as the sum of the sides is to the difference the sides, so is the tangent of half the sum of the opposite angles to the tangent of ha their difference.

It is to be observed, that in the third term of the proportion the cotangent of half tl given angle may be used instead of the tangent of the half sum of the unknown angles. Example. In the plane triangle ABC (fig. 400.),

1052. THEOREM III. When the three sides of a triangle are given.

Let fall a perpendicular from the greatest angle on the opposite side, or base, dividin it into two segments, and the whole triangle into two right-angled triangles, the propor tion will be

As the base or sum of the segments

Is to the sum of the other two sides,

So is the difference of those sides

To the difference of the segments of the base.

Then take half the difference of these segments, and add it to the half sum, or the half bas for the greater segment; and for the lesser segment subtract it.

Thus, in each of the two right-angled triangles there will be known two sides and th angle opposite to one of them, whence, by the first theorem, the other angles will be found For the rectangle under the sum and difference of the two sides is equal to the rectang under the sum and difference of the two segments. Therefore, forming the sides of thes rectangles into a proportion, their sums and differences will be found proportional. Example. In the plane triangle ABC (fig. 401.),

Let AB=345 ft.

C

So that the three angles are as follow, viz. ▲ A 27° 4′; L B 37° 20′; LC115° 36. 1053. THEOREM IV. If the triangle be right-angled, any unknown part may be found by the flowing proportion:

As radius

Is to either leg of the triangle,

So is tangent of its adjacent angle

To the other leg;

And so is secant of the same angle

To the hypothenuse.

For AB being the given leg in the right-angled triangle ABC, from the centre A with any assumed radius AD describe an arc DE, and draw DF perpendicular to AB, or parallel to BC. Now, from the definitions, DF is the tangent and AF the secant of the arc DE, or of the angle A, which is measured by that are to the radius AD. Then, because of the parallels BC, DF, we have AD; AB:: DF: BC, and :: AF: AC, which is the same as the theorem expresses in words.

Note. Radius is equal to the sine of 90°, or the tangent of 45°, and is expressed by 1 in a table of natural sines, or by 10 in logarithmic sines. Example 1. In the right-angled triangle ABC,

10.221848

2.431363

Note. There is another mode for right-angled triangles, which is as follows: ·

ABC being such a triangle, make a leg AB radius; or, in other words, from the centre A with distance AB describe an arc BF. It is evident that the other leg BC will represent the tangent and the hypothenuse AC the secant of the are BF or of the angle A.

In like manner, if BC be taken for radius, the other leg AB represents the tangent, and the hypothenuse AC the secant of the arc BG or angle C.

If the hypothenuse be made radius, then each leg will represent the sine of its opposite angle; namely, the leg AB the sine of the are AE or angle C, and the leg BC the sine of the arc CD or angle A.

G

C

P

Fig. 403.

B

D

E

Then the general rule for all such cases is, that the sides of the triangle bear to cach other the same proportion as the parts which they represent. This method is called making every side radius.

1054. If two sides of a right-angled triangle are given to find the third side, that may be found by the property of the squares of the sides (Geom. Prop. 32.; viz. That the square of the bypothenuse or longest side is equal to both the squares of the two other sides together). Thus, if the longest side be sought, it is equal to the square root of the sum of the squares of the two shorter sides; and to find one of the shorter sides, subtract one square from the other, and extract the square root of the remainder.

1055. The application of the foregoing theorems in the cases of measuring heights and distances will be obvious. It is, however, to be observed, that where we have to find the length of inaccessible lines, we must employ a line or base which can be measured, and, by means of angles, which will be furnished by the use of instruments, calculate the lengths of the other lines.

SECT. IV.

CONIC SECTIONS.

1056. The conic sections, in geometry, are those lines formed by the intersections of a pla with the surface of a cone, and which assume different forms and acquire different propert according to the several directions of such plane in respect of the axis of the cone. Th species are five in number.

1057. DEFINITIONS.-1. A plane passing through the vertex of a cone meeting the pla of the base or of the base produced is

called the directing plane. The plane VRX (fig. 404.) is the directing plane, 2. The line in which the directing plane meets the plane of the base or the plane of the base produced is called the direarix. The line RX is the directrix. 3. If a cone be cut by a plane parallel to the directing plane, the section is called a conic section, as AMB or AHI (fig. 405.)

4. If the plane of a conic section be cut by R another plane at right angles passing along the axis of the cone, the common section of the two planes is called the line of the axis.

D

5. The point or points in which the line of the axis is cut by the conic surface is or a

called the vertex or vertices of the conic section. Thus the
points A and B (figs. 404. and 405.) are both vertices, as is the
point A or vertex (fig. 406.).

o. If the line of the axis be cut in two points by the conic surface,
or by the surfaces of the two opposite cones, the portion of
the line thus intercepted is called the primary axis. The line
AB (figs. 404. and 405.) and AH (fig. 406.) is called the
primary axis.

7. If a straight line be drawn in a conic section perpendicular to the line of the axis so as to meet the curve, such straight line is called an ordinate, as PM in the above figures.

B

D

P

H

X

Fig. 406.

8. The abscissa of an ordinate is that portion of the line of axis contained between the vertex and an ordinate to that line of axis. Thus in figs. 404, 405, and 406. the parts AP, BP of the line of axis ar the abscissas AP BP.

9 If the primary axis be bisected, the bisecting point is called the centre of the coni section.

10. If the directrix fall without the base of the cone, the section made by the cutting plane is called an ellipse. Thus, in fig. 404., the section AMB is an ellipse. It is evident that, since the plane of section will cut every straight line drawn from the vertex of the cone to any point in the circumference of the base, every straight line drawn within the figure will be limited by the conic surface. Hence the axis, the ordinates, and abscissas will be terminated by the curve.

11. If the directrix fall within the base of the cone, the section made by the cutting plane is called an hyperbola. Hence it is evident, that since the directing plane passe alike through both cones, the plane of section will cut each of them, and there fore two sections will be formed. And as every straight line on the surface of the cone and on the same side of the directing plane cannot meet the cutting plane neither figure can be enclosed.

12. If the directrix touch the curve forming the base of the cone, the section made by the cutting plane is a parabola

OF THE ELLIPSIS.

1058. The primary axis of an ellipsis is called the major axis, as AB (fig. 407.); and a straight line DE drawn through its centre perpendicular to it, and terminated at each extremity by the curve, is called the minor axis.

1059. A straight line VQ drawn through the centre and terminated at each extremity by the curve is called a diameter. Hence the two axes are also diameters.

R

E M

D
Fig. 407.