17.5 CHAPTER 17. TRIGONOMETRY
�
D�
A�
B�C
h
b acd c− dIn�DCB:
a^2 = (c− d)^2 + h^2 (17.6)
from the theorem of Pythagoras.
In�ACD:
b^2 = d^2 + h^2 (17.7)
from the theorem of Pythagoras.
We can eliminate h^2 from (17.6) and (17.7) toget:
b^2 − d^2 = a^2 − (c− d)^2
a^2 = b^2 + (c^2 − 2 cd + d^2 )− d^2
= b^2 + c^2 − 2 cd + d^2 − d^2
= b^2 + c^2 − 2 cd (17.8)In order to eliminate d we look at�ACD, where we have:
cosAˆ =
d
b.
So,
d = bcosA.ˆ
Substituting this into (17.8), we get:
a^2 = b^2 + c^2 − 2 bccosAˆ (17.9)The other cases can be proved in an identical manner.
Example 15:
QUESTIONFindAˆ: