26
SPHERICAL TRIGONOMETRY.
[§47
thus
is a right angle. Therefore
; and
.
Similarly,
; therefore
therefore
.
The figure supposes
,
,
, and
each less than a right
angle; it will be found on examination that the proof will
hold when the figure is modified to meet any case which can
occur. If, for instance,
alone is greater than a right angle,
the point
will fall beyond
instead of between
and
; then
will be the supplement of
, and thus
is still equal to
.
Case III. ― Two sides, the included angle, and another angle.
48. To shew that
.
We have
Substitute the values of
and
in the first equation; thus
by transposition
divide by
; thus
49. By interchanging the letters five other formulae, like
that in the preceding Article, may be obtained; the whole
six formulae will be as follows: