107 
CHECKS ON COMPUTATIONS IN THE SOLUTION OF TRIANGLES 
A. M. KENYON 
It is the purpose of this note to illustrate methods of checking the accuracy of the 
results when unknown parts of plane triangles are computed from given parts. Five 
place tables are used in the computations. 
I. RIGHT TRIANGLES. 
Let h represent the hypotenuse and a and b the other two sides of a right triangle 
Let & be the right angle and A and B the acute angles opposite a and D respectively - 
To fix ideas suppose A is not less than B. 
Either of the following identities contains all five of the variable parts and can 
be used as a check formula when a right triangle has been completely solved. 
A R Cc 
Fig. | 
(1) 2 ab = hh? cos (A—B) 
(2) (a+b) (a—b) = h? sin (A—-B) 
To prove these produce AR (Fig, 1) to C making RC — AR, connect BC, and draw 
CD perpendicular to AB, Then CB—h and angle BCD = A—Bb, 
CD = h cos (A-B) = 20) sin A 
Therefore h?2 cos (A—-B) = 2bh sin A = 2ab 
DB = h sin (A-B) = h-2b cos A 
Therefore h? sin (A-B) = h?-2bh cos A = a?-l? 
It is evident that these formulas hold also when A is less than B. 
If it is desired to check only the sides, either of the formulas 
(3) a = (h+b) (h-b) or 6? = (h+a) (h-a) 
may be used. 
