103 Trigonometry Problems

128 103 Trigonometry Problems
(b) From the given condition and Introductory Problem 19(a), we can assume
that there is an acute triangle ABC such that
tan A 2 = x, tan B 2 = y, tan C 2 = z.
By the double-angle formulas, it suffices to prove that
tan A + tan B + tan C ≥ 3 √ 3,
which is Introductory Problem 20(b).
4. [China 1997] Let x,y,z be real numbers with x ≥ y ≥ z ≥
12 π such that
x + y + z = π 2
. Find the maximum and the minimum values of the product
cos x sin y cos z.
Solution: Let p = cos x sin y cos z. Because π 2
the product-to-sum formulas,wehave
≥ y ≥ z, sin(y − z) ≥ 0. By
p = 1 2 cos x[sin(y + z) + sin(y − z)] ≥1 2 cos x sin(y + z) = 1 2 cos2 x.
Note that x = π 2 − (y + z) ≤ π 2 − 2 · π12 = π 3
. Hence the minimum value of
p is
2 1 cos2 π 3 = 1 8 , obtained when x = π 3 and y = z = 12 π .
On the other hand, we also have
p = 1 2 cos z[sin(x + y) − sin(x − y)]≤1 2 cos2 z,
by noting that sin(x − y) ≥ 0 and sin(x + y) = cos z. Bythedouble-angle
formulas, we deduce that
p ≤ 1 4 (1 + cos 2z) ≤ 1 4
(
1 + cos π )
6
= 2 + √ 3
.
8
This maximum value is obtained if and only if x = y = 5π
24 and z = π 12 .
5. Let ABC be an acute-angled triangle, and for n = 1, 2, 3, let
x n = 2 n−3 (cos n A + cos n B + cos n C) + cos A cos B cos C.
Prove that
x 1 + x 2 + x 3 ≥ 3 2 .