PLANE TRIGONOMETRY, 
47 - 
Table II. 
Case. 
Given. 
Required. 
Solution, 
* { 
The angles 
and side A B. 
Side B C 
Side A C 
BC = AB X sin A X coscc C, 
AC^ABXsinBX cosec C. 
2 4 j | 
Two sides 
AB, B C, and 
angle C 
opposite to 
one of them. 
Angle A 
Angle B 
Side AC 
sin A =s sin C X B C -j- A B. 
B = i8o° — (A + C). 
AC = ABXsinBx cosec C. 
4&5 
Two sides 
AB, AC, 
and the 
included 
Angle A. 
Angles 
C and B 
Side B C 
B - C A 
tan —-— = (AC - AB) X cot — -5- (A C + A B). 
B + C A 
and, 2 = 90 0 — 2 : from which 
„ B + C B - C , „ B + C B-C. 
B - 2 + 2 :,njC - 2 ~ 2 
B C ~ A B X sin A X cosec C. 
6 1 
All three 
sides. 
All the 
Angles 
From half the sum of the three sides, subtract, separately, 
each of the three sides. Multiply these four numbers (the 
half sum and the three remainders) together, and take twice 
the square root of the product. This result, divided by the 
product of any two of the sides, gives the sine of the angle 
between them. 
In all plane triangles, if two of tlie angles are known, the third angle 
is found by subtracting the sum of the two from 180°. 
The foregoing equations may be solved by multiplication and division, 
with a table of natural sines, cosines, &c.; but, in order to avoid such a 
tedious process, logarithms are usually employed. In calculating with 
logarithms, multiplication is performed by adding together the log¬ 
arithms of the numbers to be multiplied: the sum is the logarithm of 
the product: division is performed by subtracting the logarithm of 
the divisor from the logarithm of the dividend; the remainder is the 
logarithm of the quotient. Twice the logarithm of a number is the 
logarithm of its square; and half its logarithm is the logarithm of its 
square root. 
The following are some of the most useful examples Qf the practice 
application of the rules given in Tables I. and II.:— 
