T R I
499
TRI
/1369) / 3)
37) .868589V..023475 V 17
74 1369 -
_— - 5
128 9785 -
111 9583
175 202
148 -
278
259
199
185
14
.023475
5
.023480 sum
add 1.255273 the log. 18.
fte sum 1.278753 is the logarithm of 19.
To find the logarithm of 2 from Series XIX. Here z—\z~ 1, and by substituting the numbers, the series becomes
V 2 = 0 +.86858896 (|+ ^ ^ to)
' V ° r k, therefore, as follows :
3 ) .86858896
9 ) .28952965 -+-
1 = .28952965
9 ) 3216996 - 5 -
3 = .1072332
9 ) 357444 -f-
5 = 71488
9 ) 39716 - 5 -
7 = 5673
9 ) 4412 -h
9 = 490
9)490 -f-
11 = 44
54 -+-
13 = 4
as the radius. Thus if we make the sides containing anyangle of equal length, and measure the width of the opening,as also the length of either side, we shall be able to find thenumber of degrees contained by the angle. Now we knowwith sufficient accuracy for all practical purposes, the num-ber of times which a radius is contained in its circumference,which is 2 X 3.14159, &c.; and by this proportion, we shallbe enabled to find our angle.
For instance, suppose the width of the opening to measure52.359 feet, and the length of the side 100 feet, then100 X 3.14159 = 314.159 gives the length of the semi-
314 159
circumference, or an arc of 180“. Then ■ ‘ - = 6 gives
0^«UOi7
the number of times which the arc to be found, is contained
180“
in an arc of 180“; therefore —— = 30“ is the measure of
6
the angle required.
The following Table gives the value of the sine, cosine, &c.in terms of each other, so that if one be given, the othermay be found.
Sin.
A =
Vi
— • cos. 2 a 1
Cos.
Ysin.
A “
A =
v 1
1
— sin. 2 A ;
— cos. A;
Tan.
sin. A ;
COS. A
Cot.
Sec.
Cosec.
A =
COS. A
sin. A
1
1
tan. A ’
A =
COS. A
1
= V 1 q. tan. *A
A ~
sin. A
^ 1 + cot S A
The following numerical values of the sine, cosine, tan-gent, and secant, of 30° 45“ 60“ 120° are of frequent use.They are calculated to radius = 1.
30°.
The chord of 60° = 2 sin. 30°
But chord of 60° = rad. = 1.
sin 30 ==
Cos. 30 = \/1 — sin. “30 = \/' i — — = — ^
] 0 , .30102996 which is the
! hay t ' lm 2 ne **rly. By only adding .00000004, we shouldt^bj 5 ‘^0103, which is the number to be found in the commonI es of logarithms. And lastly, by reverting the series,
^ + w) = m ^
— +, &c.J we obtain
1 +
AX +
th,
1.2
+
1.2.3 ‘ 1.2.3.4
+ , &c., which
e value of the number x when the logarithm of x is
Sivi
en.
Here a =
1
Tan. 30 =
sin. 30cos. 30
Sec. 30 =
1
cos. 30
1 .
2
V 3*
2
45°
Since sine of an arc = cosine of its complement $sin. 45 = cos. 90 — 45 = cos. 45.
The
Trigonometrical Formulae.
circumference of a circle being divided into 360es > of which 90“ forms a right angle, affords us a ready8 i Q Ha ^ connecting angular space with lines, for the dimen-an angle will vary directly as the arc, and inversely
But sin. s 45 -f- cos. 5 45 = 1 ;2 sin. »45 = 1; or sin. 45.
Tan. 45 =
sin. 45cos. 45
sin. 45sin. 45
V2
cos. 43.