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# A course of elementary mathematics : affording aid to candidates for admission into either of the military colleges, to applicants for appointments in the Indian civil service, and to students of mathematics generally

. (page 20 of 68)

the two perpendicular diameters BB\ DD\ are called quadrants : BD is
the first quadrant, DB' the second, JB'D' the third, and D'B the fourth.
The sine DA, of the first quadrant, is = radius ; therefore the sine of a
right angle, or sin 90= 1 : it is plain that no sine can exceed this limit;
the sines increase from sin 0, which is 0, up to sin 9Q, which is 1 ; and
they then as gradually diminish down to sin 180", which is also ; so
that the sine of any angle as the angle BAC greater than 90", is a
proper fraction, or a decimal : whatever part the linear sine CW is of the
radius DA, the same part, of course, is the numerical sine, that is, the
trigonometrical sine of the angle BAO\ of 1.

213. Cosine. The cosine of an arc, BO, is
the part Am, of the diameter BB\ which is in-
tercepted between the centre A, and m, the foot
of the sine. The numerical value of this inter-
cept, conformably to the scale radius = ], is the
trigonometrical cosine of the angle BAC. We
use the prefix trigonometrical, merely to impress
upon the mind that we are here speaking of
the sines and cosines exclusively employed in
trigonometry : we shall dispense with it in future, taking for granted that
the student will recollect that the sine or cosine of an a7igle is always an
abstract number. He may satisfy himself by a reference to Euclid as

above that the ratio r-^ is invariable, whatever be the length of the

radius AC: it is this constant ratio that-is the cosine of the angle CAB.
The sines and cosines of arcs, that is, the linear sines and cosines, do not
enter into the investigations of trigonometry.

If the angle be a right angle, its cosine is ; that is, cos 90= ; for
if C coincide with D, the intercept Ain vanishes. The cosine increases
as the angle diminishes ; for the nearer C is to B, the greater does the

TANGENT, SECANT. COTANGENT, COSECANT. 173

intercept Am become, till G coincides with B, when the angle CAB
vanishes, and the cosine becomes 1 ; that is, cos 0=1. The cosine, Am\
of an angle BAG' greater than 90, increases in an opposite direction to
Am till the obtuse angle reaches its utmost limit 180, or till G' coin-
cides with B', in which case m' also coincides with B\ so that cos 180=
1 , its linear representation lying in the opposite direction. To mark this
opposition of direction, the minus sign is always prefixed to the cosine of
an obtuse angle, so that we write cos 180= 1. In like manner, if the
angle BAG' be 150, and if the numerical value of the cosine of it be p,
we should write cos 150=: _p. This distinction of algebraic sign is im-
portant : precision requires it. In the absence of it, we could not dis-

g
cover, upon being told that the cosine of an angle A is -, whether that

angle was less or greater than 90 ; as it is, however, we know that the

3 3

statements cos A=^-, and cos ^'= -, imply two different angles, the

latter, A\ being as much greater than 90, as the former, A, is less
than 90.

214. Tangent, Secant. The tangent of an arc BCis the straight
line BT drawn from B, the origin of the arc, touching the arc at that
point, and terminating in AT, the line from the centre through the end
G of the arc. And this line, AT, is the secant of the arc BG. If the arc
exceed 90, as the arc BG\ the tangent of it BT' takes the opposite direc-
tion, which opposition is marked, as in the case of the cosine, by the minus
sign. It is easy to see that the secant continually varies its direction as
the arc varies in length : the secant of the arc BC', conformably to the
above general definition, is AT\

The ratio of the linear tangent or secant to the radius, is the tangent
or secant of the angle BA C, or BA C' : each ratio, as it is easy to see, is
invariable, whatever be the length of the radius : taking, as before, 1 for
the numerical representation of the radius, the numerical values of the
lines just defined are these ratios.

If C coincide with B, that is, if the arc be 0, the tangent BT vanishes,
and the secant AT becomes AB, the radius: .-. tan 0=0, sec 0=1.
But if C coincide with D, it is plain that AT, BT never meet, so that tan
9 **= 00, and sec 90= oo. For an obtuse angle the tangent like the
cosine is negative.

215. Cotangent, Cosecant. If a line Dt be drawn from D, the
extremity of the first quadrant, touching the arc at that point, and con-
tinued till it meet the secant of BC, or that secant prolonged, in t, the
touching line Dt is called the cotangent, and At, the cosecant of that arc.
In like manner, Dt' is the cotangent and Af the cosecant of the arc BC\
The ratio of each of these lines to the radius, or the numerical values of
the lines themselves, to the scale radius=l, are the cotangent and
cosecant of the angle BAG, or BAG'.

216. The ratios or abstract numbers to which the above six names are
given, comprehend all that are peculiar to trigonometry. There is, how-
ever, another term sometimes used : the term versed sine ; it is employed
to denote the excess of unity above the cosine of an angle ; thus vers A
means 1 cos .4. The versed sine of an arc BG is the line Bm=AB-^

174 FUNDAMENTAL TRIGONOMETRICAL RELATIONS.

Am. Also covers A means 1 sin A, In reference to the arc BG, it is
the line Dm.

to an arc or angle to make up 90, is called the complement of that arc or
angle : thus, DC is the comp. of BC, and DC^ (taken negatively) is the
comp. of BC\ Also 50 is the comp. of 40, and 50 is the comp. of
140. The cosine, cotangent, &c., of an arc or angle, are no other than
the sine, tangent, &c. of the complement of that arc or angle, regarding
the complements of arcs as measured from D.

It must be remembered then that if two arcs or angles p, q, together
make up 90, each is the complement of the other, and that therefore
sin p=cos q, cosp=sin q, tan p=cot q, cot p=sin q, &c.

218. Supplement. Whatever must be added to an arc or angle to
make up 180, is called the supplement of that arc or angle : thus CB' is
the supplement of the arc BC, and C^B' the supplement of BC\ Also
140 is the supplement of 40, 60 is the supplement of 120, and so on.
And it may here be noticed, in connection with the supplement, that
m5' is called the suversed sine of the arc BC : it is the diameter minus
the versed sine mB. Of an angle A, the suversed sine, or as it is more
frequently called, the suversine, is 2 cos A. It will have been seen,
from what has preceded, that the directive signs (+ or ) prefixed to a
cosine, tangent, or cotangent, have been suggested by the position of
the geometrical lines in the diagram. To avoid all confusion and am-
biguity about direction, the arc whose sine, cosine, tangent, &e., is to be
determined, is always considered to have its origin or commencement at
one and the same point B, and to be measured from that origin in one and
the same direction BCDC\ &c. For instance, if we had to discuss these
lines in reference to the arc CC\ we should conceive the point C applied
to B, or w^hich is the same thing, should replace CC by an arc, equal to
it, measured from B, in the direction BCD, And this is the way supple-
mental arcs are considered : the supplement of BC is CB\ as to length,
or number of degrees; but to determine its sine, cosine, &c., we conceive
C to be brought down to B, and the arc, thus displaced, to terminate at C\
so that (7W is the sine of that supplement, and Am' its cosine. As a
supplement is what the arc wants of 180, or of a semicircumference, it is
plain that if BC is equal to CB, we must have BC=B'C\ and Cm=
C'm\ as also Am=Am' : hence an arc or angle has the same sine as the
supplement of that arc or angle : the cosine too is the same in magnitude
for both, but the signs of direction (+ or ) are opposite : thus, if sin
40=p, then also, sin 140=^; but if cos 46=^', then cos 140= q.
The signs of direction prefixed to the trigonometrical ratios, thus have their
origin in the linear representation of those ratios : in a diagram, fixing
direction as well as length : we could not with propriety speak of a ratio,
or abstract number, taking a directive sign, uidess that ratio or number
were derived from geometrical considerations. The lines in the diagram
at (213), and which have been defined above, may always be regarded as
linear representations of the trigonometrical ratios bearing the same
names, the radius representing 1 ; so that many properties and relations
of these ratios may be established by help of their representative lines, as
in the following article.

219. Fundamental Trigonometrical Relations. Refer-
ring to the diagram at art. (213), and remembering that the trigonometrical

FUNDAMENTAL TEIGONOMETEICAL EELATIONS. 175

radius is 1, the 47th prop, of Euclid's first book furnishes the following-
relations, where, for simplicity of notation, (sin Af, (cos A)^, &c. are
written sin^ A, cos^ A, &c.
siii2^+cosM=l2 .-. sin J=^/(l-cos2^), cos^=x/(l-sin2^) \

l2+tanM=sec2 A ,'. sec ^=v^(l+tan2^), tan 4 = ^(sec^ ^-1) [ [1].

l^+cot^ A=icosec^ A .'. cosec -4=^^(1 4-cot^ A), cot ^=y^(cosec2 A1))

Again : since the sides about the equal angles of equiangular triangles
are proportional (Euc. Prop. 4, B. VI.), by comparing together the equi-
angular triangles TAB, CAm, and the equiangular triangles fAD, C'Am\
where Am!^=.Am, and Cmf-=^Cm, we have the relations

tan A sin A cot A cos ^ cos ^ 1 sin ^ 1

1 cos J.' 1 sin A^ 1 sec J.' 1 cosec A^

sin ^ cos ^ 1 1

that IS, tan A=z -, cot A=z -, sec i[= -, cosec Az=:~ -

cos A sin J. cos A sin J.

[2].

The following pairs of quantities therefore, namely, tan A, cot A ; cos
A, sec A ; sin A, cosec A, are the reciprocals of each other; and this it
will be necessary to recollect.

The foregoing fundamental relations show that from the two trigono-
metrical quantities sin A, cos A, all the others may be derived, as well as
regards sign, as numerical value : we thus see, for instance, that the sign
of the secant of an angle is the same as that of the cosine, and that the
sign of the cosecant, is the same as that of the sine.

220. All the quantities in the preceding equations are numbers : the
trigonometrical ratios : they apply exclusively to angles. But they may
easily be converted into geometrical relations, applicable to the sines,
cosines, &c., of arcs belonging to any circle. If we write the terms
SINE, COSINE, &c., in capitals, when we refer to arcs instead of angles,

we have only to replace sin A, cos A, &c., above, by 5, 75, &c.,

where B, represents the radius of the circle to which the arc A belongs.
And to this R we may, of course, afterwards attribute any numerical value
we please. We shall now give an example or two in reference to the re-
lations marked [1] and [2] above.

(1) Required the sine of 45. Here ^=45o=90-45: hence (217)

sin ^=cos A .'. [1], sin'^ -4-j-cos^ .4=2 sin^ A=l

.-. sin ^=^^- = -^2=cos A.

(2) Required the tangent, cotangent, &c., of 45. Dividing the sine by

the cosine, we have [2], tan 45=-\/2-r - 'v/2=l, also cot 45= 1 ; and

2 2

since the secant is the reciprocal of the cosine, and the cosecant of the
sine, we have sec 45=1-h-v'2='/2, and cosec 45=v/2.

(3) Required the sine, cosine, &c., of 60. This is the measure of each
angle of an equilateral triangle, since the three equal angles amount
together to two right angles, or 180. If from the vertex of such a
triangle a perp. be drawn, it will bisect both the base and the vertical
angle, dividing the equil. triangle into two equal right-angled triangles.

1 76 TABLES.

Let CAC\ in the diagram at p. 171, be an equil. triangle, CC being
equal to the radius of the circle passing through C, C" : then ^4C=30,

therefore to radius AB=J, Am=-=sui 30% .*. [1],
coa30''=v'(l-^)=^N/3.

Also[2], tan30'=i||i:=i34v3, cot 30-^3,

12 2 1

sec 30= -= -=-^3, cosec 30=-^ =2.

cos 30 V3 3^ sin 30

Since the complement of 30 is GO*", we infer from these values (217) that
8in30''=cos 60"=-, cos 30"=sin 60''=- -v/ 3, tan 30=cot 60''=-'v/3,

li ti O

2
cot 30"'=tan 60''= v/ 3, sec 30''=cosec 60=- ^3, cosec 30=sec 60=2.

o

221. Tables. But it would not be possible, in this way, to deter-
mine the trigonometrical ratios for all values of the angle A. For this
general purpose, certain series have been investigated, and certain alge-
braic expedients devised, which will be explained hereafter. With such
aid, the sines, cosines, &c., for all values of A, from A=i), up to ^=90,
have been computed to several places of decimals, and arranged in Tables.
For values of A above 90", such computations would be superfluous, for
the sine, cosine, &c., of an angle above 90, are the same, in numerical
value, as the sine, cosine, &c., of an angle as much below 90. A mere
inspection of the diagram at p. 171 is sufficient to show this.

There are two kinds of trigonometrical tables. The first contains the
several trigonometrical ratios, computed as above hinted. This is called
a table of natural sines, cosines, &c. The second is a table of logarithmic
sines, cosines, &c. These are not strictly the logs of the natural sines,
cosines, &c. : they are these logs each increased by the number 10, which
addition to them is made solely to avoid the inconvenience of negative
logs. The trigonometrical sines and cosines, as we have already seen, are all
less than 1 ; so that the logs of these are all negative : by increasing each
log by 10, we in effect compute the table to the radius jR=10^'^, instead
of to 72=1. And we have seen (220) that the trigonometrical ratios
remain the same whatever be R.

222. Before concluding these introductory observations, there is one
other inference, from the diagram at p. 171, to which the attention of
the student must be directed. The chord CG" of an arc GBC" is
obviously double the sine Cm of half that arc ; and still regarding the
circle to which the arc belongs as the trigonometrical circle or the circle
whose radius is the linear representation of the numerical unit we infer
that the chord of any angle is twice the sine of half that angle, or that

did ^=2 sin - A, Now by Euc. 8, VI., the chord CB is a mean pro-

portional between BB' , Bm ; that is, CB^^BB' x Bm,, giving to the lines
their numerical values, but BB'=^,

.'. 2(1 cos A)=:2 vers ^=(2 sin -Ay .'. vers A=2 sin' -A,
u 2

EIGHT-ANGLED TRIANGLES.

177

a property that will be called in request in investigating the third case of
oblique-angled triangles.

Having thus disposed of the necessary preliminaries, we may now pro-
ceed to the business of Trigonometry Proper.

Paet 1. Solution of Plane Triangles.

2Q3. Right-angled Triangles. There are two cases to be con-
sidered : 1. When an acute angle and a side are given. Eule I. With
the vertex of the given angle as centre, and the given side, terminating in
that vertex, as radius, imagine an arc to be described, and notice what
trigonometrical name, in reference to that arc whether sine, or cosine, or
tangent, &c., the sought side takes.

2. Enter the table of natural sines, &c., with the given angle, and take
out the number under that name. Multiply the given side by that
number : the product will be the sought side.

[Note. When one of the acute angles of a right-angled triangle is
given, the other is virtually given : for if A be one, 90" A is the other].

2. When two sides are given to find an angle. Rule 1. With the
vertex of the required angle as centre, and a given side, terminating in
that vertex, as radius, imagine an arc to be described, and notice what
trigonometrical name the other given side takes.

2. Divide this latter by the former : the quotient will be the tabular
number of the same name, over which, in the table, the value of the
required angle will be found.

For, in the right-angled triangle ABC, let, first,
the side AB=c, and the angle A be given : then
if with centre A, and radius AB, we imagine an
arc to be described, as in the diagram, we see that
BC or a becomes the tangent of that arc, and AC
or b, the secant ; and we know, whatever be the
length of our radius, that we have the constant

ratio -= tan A, and -=sec A. Next, let the hypo-
c c '^

tenuse AC=b, and the angle A, be given : then if

with centre A, and radius AC, we imagine an arc

to be described, it is plain that BC or a becomes

the sine of that arc, and AB or c the cosine ; and

we know, whatever be the length of our radius, that we have the constant

a c

ratio -=8in A, and 7=co8 A, .-. a=c tan A, b=c sec A, a=b sin A, c=

b cos A, and

A ^ A ^ A ^ A ^

tan A =-, sec A =-, sm A =-, cos A =-.
c c b b

results which sufficiently establish the foregoing rules. From the expres-

/J
sions for tan A, and sin A, and from (219), it is plain that cot A=-,

and cosec A~- ; and therefore that G=a cot A, and b=:a cosec A.
a

Note 1. The following examples, on the solution of right-angled tri-
angles, are all worked by the table of natural sines, cosines, &c. In the

N

178 RIGHT-ANGLED TEIANGLES.

case of right-angled triangles, the operations are, in general, easier by the
natural numbers than by logarithms. Prefixed to the tables will be found
all the necessary directions for using them. In the common applications
of plane trigonometry it is but seldom requisite to compute the angles
nearer than to degrees and minutes.

2. In the table of natural sines, &c., the secants and cosecants are not

inserted: for since sec A= -, and cosec A=- , we may change

cos A sin A

multiplication by sec A, or by cosec A, for division by cos A, or by sin A ;

and division by sec A, or by cosec A, for multiplication by cos A, or by

sin A. In all multiplications and divisions by large numbers, where

several decimals are concerned, it is better to use the contracted methods,

as explained in the Arithmetic. (See VVeale's Eudimentary Treatise.)

When any two sides of a right-angled triangle are given to find the
third side, the solution is effected by the 47th of Euclid's first book ; and
no aid from trigonometry is required. Putting for the sides opposite to
the angles A, B, C, the corresponding small letters a, b, c, if B be the
right angle, we have, by the prop, referred to,

62=a2_|.c2 ... J=^(a2+c2), a=^{b^-c^)=z^{{b-\-c){h-c)}, c=^ {{b+a){h~a)}.

(1) In the right-angled triangle ABC, are given c=174 feet, and A=
IS** 19', to find the sides a and b.

Here a=c tan A, and b=c sec A=c-^cos A (art. 219)

tan 18 19'= -33104 cos 18 19'= 9,4,9,3,3) 174 (183-29=6
c, reversed, = 471 94933

33104 79067

23173 75946

1324

3121

a=57-601 2848

273

190

83

Hence the side 5C=57-6 feet, and the side ^C=183-29 feet.

(2) From the edge of a ditch 18 feet wide, which surrounded a fort, the
angle of elevation of the top of the wall was taken, and found to be
62" 40': required the height of the wall, and the length of a ladder
necessary to scale it. Here c=18 feet, and ^ = 62" 40'. And tan
62 40' X 18 = 1-93470 X 18 = 34-82 feet=a, the height of the wall,
18-Hcos 62-^ 40' = 18-J - 459 17 = 39-2 feet=6, the length of the ladder.

(3) The hypotenuse ^C is 643-7 feet, and the base AB, 473-8 feet:

AB
required the angles A, C, and the perpendicular, BC. Here =cos A,

that is, 473-8-T-643-7=-73606=cos 42" 36' .-. ^=42" 36', and C=90"
^=47" 24'. Also (Euc. 47 of I.), s/ (643-7^ 473-8'0=435-87 feet=^C.

(4) The tower DC oi an enemy's fortress cannot be safely approached
nearer than B : at this point the angle of elevation CBD of the top is
found to be 55" 54'. The observer, upon going back a further distance
BA of 100 feet, finds the angle of elevation A to be 33" 20' : required the
height of the tower and its distance from B.

RIGHT-ANGLED TRIANGLES.

179

DA=CD tan ACD, DB=CD tan BCD, that is,
DA=CD cot A, DB=CD cot CB2) .'. AB=DA
DB=CD{cot 33" 20' cot 55 54')=CD (1-52043
67705)=100 .-. (7D=100-^ 84338=118-57 ft.,
the height of the tower. And ..CD cot CBD=
118-57 X 67705=80-28 ft., the distance BD.

(5) From the top of a mountain BA, m
miles high, the angle of depression EAG,
below the horizontal line AE, of the re-
motest visible point of the surface of the
sea, is taken, and found to be : it is re-
quired to determine the radius OG of the
earth.

Since OAE:=Q0\ EAC is the comple-
ment of OAC, but O is also the complement
of OAC .-. 0=EAC=Q.

Now OA = OC sec 9=0B + BA

,'.BA = m = OC sec Q-OC=OC{secQ-^l).:OC=

m

sec 01

or, smce

sec 0=l-i.cos e, DC-

m cos 6

miles.

1 COS0

Note. This would be a very expeditious method of determining the
earth's diameter, if the refraction of the atmosphere, near the horizon,
could be accurately estimated. But owing to the fluctuations of this
horizontal refraction, it is of uncertain amount; so that the foregoing
method cannot be regarded as giving more than a somewhat rough
approximation to the truth.

The radius of the earth being known, however, from other investiga-
tions, it is easy to find sufficiently near for the purpose, and without the
aid of trigonometry at what distance at sea the top of a mountain of
known height just emerges above the distant horizon. For, calling the
height h, the distance d, and the radius of the earth R, we know by
Euclid, Prop. 36, B. III., that {2R-^h)h=(P. As h" is comparatively
insignificant, we may neglect it : we shall then have d'=:U.Rh. Thus,
R being about 4000 miles, if /i be 1 mile, 6^=^8000=89 miles. Also,
the distance being known, we may compute the height of the mountain

from /i=- ; thus, if a ship just loses sight of the top of a mountain,

/iJtX

when at the distance of 30 miles from it, its height will be

:594 feet.

900 9 .,
= - miles

8000 80

Examples for Exercise.

(1) The base of a right-angled triangle is 246 feet, and the angle at the
base 33 45' : required the perpendicular and hypotenuse.

(2) The hypotenuse is 430 yards, the perpendicular 214 yards : required
the angle between them.

(3) Given c=73, ^=52 34' to find a, b.

(4) Given c=288, ^=63 8' to find a, b.

(5) Given a=85, J=lll-4 to find A, c.

(6) Given a=75-18, c=53-42find5, A, C.

(7) Given c=138-24, 0=52 57' find A.

(8) The perp. is double the base : required

the angles.

N 2

180 OBLIQUE-ANGLED TRIANGLES.

(9) A tower 53J feet high, is surrounded by a ditch 40 feet broad : the
angle of the elevation of the top of the tower from the opposite bank is
53'' 13' : required the length of a ladder sufficient to scale the tower.

(10) If the top of a mountain, known to be 6600 feet high, can just be
seen at sea 100 miles off, what must be the diameter of the earth ?

(1 1) From the top of a ship's mast, 80 feet above the water, the angle
of depression of another ship's hull was found to be '20: required the
distance between the two ships.

(12) A ladder, 40 feet long, is so placed as to reach a window 33 feet
from the ground ; and upon being turned over, the foot remaining in the
same place, it reaches a window 2 1 feet from the ground, on the opposite
side of the street : required the breadth of the street.

(1 3) From the top of a castle standing on a hill by the sea- side, the
angle of depression of a ship at anchor was observed to be 4 52'; at the
bottom of the castle, or top of the hill, the angle of depression was 4" 2' :
required the horizontal distance of the ship, as also the height of the hill,
that of the castle itself being 60 feet.

(14) The height of the mountain called the Peak of Teneriffe is about
2^ miles ; the angle of depression of the remotest visible point of the
surface of the sea is found to be 1 58' : required the diameter of the
earth, and the utmost distance at which an object can be seen from the
top of the mountain that is, the length of the line from the eye to the
remotest visible point.

(15) The angles of elevation of a balloon were taken by two observers
at the same time ; both were in the same vertical plane as the balloon,
and on the same side of it ; the angles were 35" and 64, and the observers
were 880 yards apart : required the height of the balloon.

224. Oblique-angled Triangles. Every triangle has six parts
as they are called three sides and three angles. If any three of the
six be given, except they be the three angles, the remaining three may
be determined by computation. It is plain that the three angles would
not suffice for the determination of the triangle, because there may be an
infinite variety of equiangular triangles all different in size. There are
three cases to be considered.

225. Case I. When two of the three given parts are opposite parts an
angle and a side.

 Using the text of ebook A course of elementary mathematics : affording aid to candidates for admission into either of the military colleges, to applicants for appointments in the Indian civil service, and to students of mathematics generally by John Radford Young active link like:read the ebook A course of elementary mathematics : affording aid to candidates for admission into either of the military colleges, to applicants for appointments in the Indian civil service, and to students of mathematics generally is obligatory