Plane trigonometry, by S.L. Loney.

216

TRIGONOMETRY.

[Exs. XXXIII.]

19. A flagstaff PN stands on level ground. A base AB is measured
at right angles to AN, the points A, B and N being in the same horizontal
plane, and the angles PAN and PBN are found to be a and a respectively.
Prove that the height of the flagstaff is
A sin a sin /,/sin (a - p) sin (a + p)
If AB = 100 feet, a= 70~, and 3= 50~, calculate the height.
20. A man standing due south of a tower on a horizontal plane
through its foot finds the elevation of the top of the tower to be 54~ 16';
he goes east 100 yards and finds the elevation to be then 50~ 8'. Find
the height of the tower.
21. A man in a balloon observes that the angle of depression of an
object on the ground bearing due north is 33~; the balloon drifts 3 miles
due west and the angle of depression is now found to be 21~. Find the
height of the balloon.
22. From the extremities of a horizontal base-line AB, whose length
is 1000 feet, the bearings of the foot C of a tower are observed and it is
found that z CAB=56~23', CBA=47~15', and that the elevation of
the tower from A is 9~ 25'; find the height of the tower.
196. Ex. A flagstaff is on the top of a tower which
stands on a horizontal plane. A person observes the angles,
a and /3, subtended at a point on the horizontal plane by the
flagstaff and the tower; he then walks a known distance a
toward the tower and finds that the flagstaff subtends the
same angle as before; prove that the height of the tower
and the length of the flagstaff are respectively
a sin / cos (a + 8) a sin a
cos (a + 2/3) cos (a + 2/3)
Let P and Q be the top and foot of the tower, and let
PR be the flagstaff. Let A and B be the points at which
the measurements are taken, so that z PA Q = and
Z PAR = Z PBR = a. Since the two latter angles are
equal, a circle will go through the four points A, B, P,
and R.