Plane trigonometry, by S.L. Loney.

186

TRIGONOMETRY.

[Exs. XXVILI]

17. The sides of a right-angled triangle are 21 and 28 feet; find the
length of the perpendicular drawn to the hypothenuse from the right
angle.
18. If in any triangle the angles be to one another as 1:2:3,
prove that the corresponding sides are as 1: /3: 2.
19. In any triangle if
A 5 B 20
tan-l tan =,
2 6' 2 - 37'
C
find tan, and prove that in this triangle a +c = 2b.
20. In an isosceles right-angled triangle a straight line is drawn from
the middle point of one of the equal sides to the opposite angle. Shew
that it divides the angle into parts whose cotangents are 2 and 3.
21. The perpendicular AD to the base of a triangle ABC divides it
into segments such that BD, CD and AD are in the ratio of 2, 3 and 6;
prove that the vertical angle of the triangle is 45~.
22. A ring, ten inches in diameter, is suspended from a point one
foot above its centre by 6 equal strings attached to its circumference at
equal intervals. Find the cosine of the angle between consecutive
strings.
23. If a2, b2 and c2 be in A.P., prove that cot A, cot B and cot C are in
A.P. also.
A B
24. If a, b and c be in A.P., prove that cos A cot, cos B cot-
2i 2
C
and cos C cot - are in A. P.
2
A B C
25. If a, b and c are in H.P. prove that sin2, sin2 and sin2 are
also in H.P.
26. The sides of a triangle are in A.P. and the greatest and least
angles are 0 and 0; prove that
4 (1 - cos 0) (1 - cos 0) = cos 0 + cos q.
27. The sides of a triangle are in A.P. and the greatest angle exceeds
the least by 90~; prove that the sides are proportional to /7 + 1,,/7 and
J7-1.