103 Trigonometry Problems

3. Advanced Problems 75
9. Find the minimum value of
| sin x + cos x + tan x + cot x + sec x + csc x|
for real numbers x.
10. Two real sequences x 1 ,x 2 ,... and y 1 ,y 2 ,... are defined in the following
way:
x 1 = y 1 = √ √
3, x n+1 = x n + 1 + xn 2, y y n
n+1 =
1 + √ ,
1 + yn
2
for all n ≥ 1. Prove that 2 1.
11. Let a,b,c be real numbers such that
sin a + sin b + sin c ≥ 3 2 .
Prove that
(
sin a − π ) (
+ sin b − π ) (
+ sin c − π )
≥ 0.
6
6
6
[ √ √ √ √ ]
12. Consider any four numbers in the interval 2− 6
2
, 2+ 6
2
. Prove that there
are two of them, say a and b, such that
∣
√ √ ∣ ∣∣
∣a 4 − b 2 − b 4 − a 2 ≤ 2.
13. Let a and b be real numbers in the interval [0, π 2
]. Prove that
if and only if a = b.
sin 6 a + 3 sin 2 a cos 2 b + cos 6 b = 1
14. Let x,y,z be real numbers with 0