ELEMENTS

OF

PLANE TRIGONOMETRY.

TRIGONOMETRY is the application of Arithmetic to Geometry: or, more precisely, it is the application of number to express the relations of the sides and angles of triangles to one another. It therefore necessarily supposes the elementary operations of arithmetic to be understood, and it borrows from that science several of the signs or characters which peculiarly belong to it.

The elements of Plane Trigonometry, as laid down here, are divided into three sections: the first explains the principles; the second delivers the rules of calculation; the third contains the construction of trigonometrical tables, together with the investigation of some theorems, useful for extending trigonometry to the solution of the more difficult problems

SECTION I.

LEMMA I.

An angle at the centre of a circle is to four right angles as the arc on which it stands is to the whole circumference.

Let ABC be an angle at the centre of the circle ACF, standing on the circumference AC: the angle ABC is to four right angles as the arc AC to the whole circumference ACF.

Produce AB till it meet the circle in E, and draw DBF perpendicular to

AE.

Then, because ABC, ABD are two angles at the centre of the circle ACF, the angle ABC is to the angle ABD as the arc AC to the arc AD, (33. 6.); and therefore also, the angle ABC is to four times the angle ABD as the arc AC to four times the arc AD (4. 5.).

But ABD is a right angle, and therefore four times the arc AD is equal to

D

the whole circumference ACF; therefore the angle ABC is to four right angles as the arc AC to the whole circumference ACF.

COR. Equal angles at the centres of different circles stand on arcs which have the same ratio to their circumferences. For, if the angle ABC, at the centre of the circles ACE, GHK, stand on the arcs AC, GH, AC is to the whole circumference of the circle ACE, as the angle ABC o four right angles; and the arc HG is to the whole circumference of the circle GHK in the same ratio.

DEFINITIONS.

1. Ir two straight lines intersect one another in the centre of a circle, the rc of the circumference intercepted between them is called the Measure of the angle which they contain. Thus the arc AC is the measure of the angle ABC.

2. If the circumference of a circle be divided into 360 equal parts, each of these parts is called a Degree; and if a degree be divided into 60 equal parts, each of these is called a Minute; and if a minute be divided into 60 equal parts, each of them is called a Second, and so on. And as many degrees, minutes, seconds, &c. as are in any arc, so many degrees, minutes, seconds, &c. are said to be in the angle measured by that arc.

COR. 1. Any arc is to the whole circumference of which it is a part, as the number of degrees, and parts of a degree contained in it is to the number 360. And any angle is to four right angles as the number of degrees and parts of a degree in the arc, which is the measure of that angle, is to 360.

COR. 2. Hence also, the arcs which measure the same angle, whatever be the radii with which they are described, contain the same number of degrees, and parts of a degree. For the number of degrees and parts of a degree contained in each of these arcs has the same ratio to the number 360, that the angle which they measure has to four right angles (Cor. Lem. 1.).

The degrees, minutes, seconds, &c. contained in any arc or angle, are usually written as in this example, 49°. 36′. 24′′. 42""; that is, 49 degrees, 36 minutes, 24 seconds, and 42 thirds.

3. Two angles, which are together equal to two right angles, or two arcs which are together equal to a semicircle, are called the Supplements of one another.

4. A straight line CD drawn through C, one of the extremities of the arc

5. The segment DA of the diameter passing through A, one extremity of the arc AC, between the sine CD and the point A, is called the Versed sine of the arc AC, or of the angle ABC.

6. A straight line AE touching the circle at A, one extremity of the arc AC, and meeting the diameter BC, which passes through C the other extremity, is called the Tangent of the arc AC, or of the angle ABC

COR. The tangent of half a right angle is equal to the radius.

7. The straight line BE, between the centre and the extremity of the tan gent AE is called the Secant of the arc AC, or of the angle ABC.

COR. to Def. 4, 6, 7, the sine, tangent and secant of any angle ABC, are likewise the sine, tangent, and secant of its supplement CBF.

It is manifest, from Def. 4. that CD is the sine of the angle CBF. Let CB be produced till it meet the circle again in I; and it is also manifest, that AE is the tangent, and BE the secant, of the angle ABI, or CBF, from Def. 6. 7.

COR. to Def. 4, 5, 6, 7. The sine, versed sine, tangent, and secant of an

B

E

P

OMD

A

Let AC, MN be measures of the angle ABC, according to Def. 1.; CD the sine, DA the versed sine. AE the tangent, and BE the secant of the arc AC, according to Def. 4, 5, 6, 7 NO the sine, OM the versed sine, MP the tangent, and BP the secant of the arc MN, according to the same definitions. Since CD, NO, AE, MP are parallel, CD: NO :: rad. CB : rad. NB, and AE: MP :: rad. AB: rad. BM, also BE: BP :: AB: BM; likewise because BC: BD :: BN: BO, that is, BA: BD :: BM: BO, by conversion and alternation, AD: MO:: AB: MB. Hence the corollary is manifest. And

therefore, if tables be constructed, exhibiting in numbers the sines, tangents secants, and versed sines of certain angles to a given radius, they will exhibit the ratios of the sines, tangents, &c. of the same angles to any radius whatsoever.

In such tables, which are called Trigonometrical Tables, the radius is either supposed 1, or some in the series 10, 100, 1000, &c. and construction of these tables are about to be explained.

8. The difference between any angle

and a right angle, or between any arc and a quadrant, is called the Complement of that angle, or of that arc. Thus, if BH be perpendicular to AB, the angle CBH is the complement of the angle ABC, and the arc HC the complement of AC; also, the complement of the obtuse angle FBC is the angle HBC, its excess above a right angle; and the complement of the arc FC is HC.

F

The use

H

L

E

A

B

D

9. The sine, tangent, or secant of the complement of any angle is called the Cosine, Cotangent, or Cosecant of that angle. Thus, let CL or DB, which is equal to CL, be the sine of the angle CBH; HK the tangent, and BK the secant of the same angle: CL or BD is the cosine, HK the cotangent, and BK the cosecant of the angle ABC.

COR. 1. The radius is a mean proportional between the tangent and the cotangent of any angle ABC; that is, tan. ABC × cot. ABC=R2. For, since HK, BA are parallel, the angles HKB, ABC are equal, and KHB, BAE are right angles; therefore the triangles BAE, KHB are similar, and therefore AE is to AB, as BH or BA to HK.

COR. 2. The radius is a mean proportional between the cosine and secant of any angle ABC; or

cos. ABCX sec. ABC-R?.

Since CD, AE are parallel, BD is to BC or BA, as BA to BE.

PROP. I.

In a right angled plane triangle, as the hypotenuse to either of the sides, so the radius to the sine of the angle opposite to that side; and as either of the sides is to the other side, so is the radius to the tangent of the angle opposite to that side.

Let ABC be a right angled plane triangle, of which BC is the hypotenuse. From the centre C, with any radius CD, describe the arc DE; draw DF at right angles to CE, and from E draw EG touching the circle in E, and meeting CB in G; DF is the sine, and EG the tangent of the arc DE, or of the angle C.

The two triangles DFC, BAC, are equiangular, because the angles DFC, BAC are right angles, and the angle at C is common. Therefore, CB: BA :: CD: DF; but CD is the radius, and DF the sine of the angle C, (Def. 4.); therefore CB: BAR sin. C.

:

Also, because EG touches the circle in E, CEG is a right angle, and therefore equal to the angle BAC; and since the angle at C is common

G

Ꭰ

to the triangles CBA, CGE, these triangles are equiangular, wherefore CA: AB:: CE: EG; but CE is the radius, and EG the tangent of the angle C; therefore, CA: AB:: R: tan. C.

COR. 1. As the radius to the secant of the angle C, so is the side adjacent to that angle to the hypotenuse. For CG is the secant of the angle G (def. 7.), and the triangles CGE, CBA being equiangular, CA: CB:: CE: CG, that is, CA: CB:: R: sec. C.

COR. 2. If the analogies in this proposition, and in the above corollary be arithmetically expressed, making the radius = 1, they give sin. C = BC; tan. C

AB

AB
AC'

sec. C =

BC
AC

Also, since sin. C=cos. B, because B

is the complement of C, cos. B =

AB
BC'

and for the same reason, cos. C =

AC

BC

COR. 3. In every triangle, if a perpendicular be drawn from any of the

angles on the opposite side, the segments of

that side are to one another as the tangents of
the parts into which the opposite angle is di-
vided by the perpendicular. For, if in the tri-
angle ABC, AD be drawn perpendicular to
the base BC, each of the triangles CAD, ABD
being right angled, AD: DC: R: tan. CAD,
and AD: DB:: R: tan. DAB; therefore, ex B
æquo, DC: DB:: tan. CAD: tan. BAD.

SCHOLIUM.

A

D

The proposition, just demonstrated, is most easily remembered, by stating thus: If in a right angled triangle the hypotenuse be made the radius, the sides become the sines of the opposite angles; and if one of the sides be made the radius, the other side becomes the tangent of the opposite angle, and the hypotenuse the secant of it.