Plane trigonometry, by S.L. Loney.

HEIGHTS AND DISTANCES.

217

To get the height of the flagstaff we have to connect
the unknown length PR with the known length AB.
This may be done by connecting each with the
length AR.
To do this, we must first /
determine the angles of the /
triangles ARP and ARB.
Since A, B, P, and R lie on a
circle, we have
BRP = BAP = /3,
and Z APB = ARB = 0 (say). A B Q
Also z APR = 90~ + z PAQ = 90~ +.
Hence, since the angles of the triangle APR are
together equal to two right angles, we have
so0~= + (90~ + 8) + (e +/3),
so that = 90~ - (a + 23)................. (1).
From the triangles APR and ABR we then have
PR AP AAR a
sin ca sin RPA sin RBA sin (Art. 163).
[It will be found in Chap. XV. that each of these
quantities is equal to the radius of the circle.]
Hence the height of the flagstaff
a sin a a sin a
sin 0 cos(a + 2/') y
PQ
Again PB = cos BPQ = cos (a + f)............(2),
PB sin PAB sin 3
and......... (3).
a sin APB sin 0....
Hence, from (2) and (3), by multiplication,
PQ _ sin / cos (a + /) sin 3 cos (a /) b ()
a sin 0 cos (a + 2/,) '