TRIGONOMETRY. 
The side B C 109 
BD _7^ 
Sum... 1*55 DilF.. 
To find the angles D and C. 
As the sum of the sides B C and B D — 135 2.56717 
Is to the difference 33 1.51851 
So is tire tangent of half the sum of angles C and D 39° 15' 9.91224 
11.43075 
2.26717 
To the tangent of half the difference of the angles C and D 8° 17' 9.16358 
To half the sum of the angles D and C 39° 15' 
Add half the difference of the angles C and D 8" I?' 
Gives the greater angle D .....' 47“ 32' 
But if subtracted (from 39“ 15' ) gives the lesser angle 30" 58' 
109 180" 0' 
76 101° 30' 
33 78“ 30' sum of the two angles 
Half.... 39“ 1 . 5 ' atDandC. 
•Having the two angles, the side is found according to Axiom II: for it will be. 
To find D C. 
As the sine angle D 47° 32' 9.86736 
Is to the sine B C 109 2.03743 
So is sine angle B 101 . 30u 9.99116 
12.02862 
To the side D C required 144.8 
Axiom IV, In any plane triangle, as the 
base, or greater side, is to the sum of the 
other two sides ; so is the difference of the 
sides to the difference of the segments of 
the base, made by a perpendicular let fall 
from the angle opposite to the base : and if 
half the difference of the segments be add- 
9.86786 
2.16076 
ed to half their sum, it will give the greater 
segment ; but if subtracted, the remainder 
will be the lesser segment. The triangle 
being thus cut, becomes two right angled 
triangles ; the hypothenuses and bases of 
which are given to find the angles by 
Axiom 1. 
Three sides given to find the angles. 
'The side B C 105, B D 85, and C D 50 miles, being given to find the angles B D C, 
BCD, andCBD, fig. 5. 
B D = 85 
I CD = 50 
The sum of the two shortest sides j35 
The difference of them 35 
The proportions will be 
As the side B C; 105 — 2.02119 — 52i the half of greatside. 
Is to the sum of the sides BD and DC 135 — 2.13033 — 22| half d:ff. of segment. 
So is the diff. of the' sides BD and DC 35 — 1.54417 greatest segment, 
3.67440 
, 2.02119 30 the lesser segment. 
Difference of the segment of the base, 
or great side 
451.65321 
I 
