103 Trigonometry Problems

17. Prove that
4. Solutions to Introductory Problems 93
(
1 + a ) ( 1 + b ) (
≥ 1 + √ ) 2
2ab
sin x cos x
for all real numbers a,b,x with a,b ≥ 0 and 0 1 + 2ab + 2√ 2ab.
By the arithmetic–geometric means inequality, we obtain
a
sin x +
b
cos x ≥
2 √ ab
√
sin x cos x
.
By the double-angle formulas,wehavesinx cos x =
2 1 sin 2x ≤ 2 1 , and so
2 √ ab
√
sin x cos x
≥ 2 √ 2ab
and
ab
sin x cos x ≥ 2ab.
Combining the last three inequalities gives the the desired result.
18. In triangle ABC, sin A + sin B + sin C ≤ 1. Prove that
min{A + B,B + C, C + A} < 30 ◦ .
Solution: Without loss of generality, we assume that A ≥ B ≥ C. We
need to prove that B + Ca) imply that sin B+sin C>sin A,sosinA+sin B+sin C>2 sin A.
It follows that sin A<
2 1 A+B+C
, and the inequality A ≥
3
= 60 ◦ gives that
A>150 ◦ ; that is, B + C