Plane trigonometry, by S.L. Loney.

472

TRIGONOMETRY.

EXAMPLES. LXVIII.
1. If x + y be a given angle, less than 7r, prove that
(1) sin x + siny, (2) cos2x+cos2y, and (3) cosx cosy
all have their greatest values when x=y.
2. Find the minimum value of
a2 tan x + b2 cot x.
Find the minimum values of
2cos 2 c3
3. 23 + 4. a2sin2 + b2cosec2 0.
2 cos0O'
5. If x+y=a, where a is:> 2, find when tan x tan y is a maximum.
2 cos a
We have 1- tan x tan y = 2 cos a- 2)
cos a - cos (a - 2x)
6. Prove that the maximum triangle having a given perimeter is
equilateral.
tA tan
The area of a triangle can be proved to equal s2 tan tan 2 tan2
7. Prove that the area of the pedal triangle of an acute-angled
triangle is never greater than one quarter of the area of the latter.
8. If ABC be a triangle, prove that the least value of
3
cos 2A + cos 2B + cos 2C is -.
Prove also that cos A +cos B + cos C is always >1 and not greater
than 3
On the geometrical representation of complex
quantities.
401. In Chap. IV. we pointed out that if a distance
in any direction (say, horizontally towards the right) be
represented by a, then -a represents the same distance
drawn in an opposite direction, i.e. horizontally towards
the left.