Fear of flying persistent fear or anxiety when they consider flying, as well as during flying. They will avoid flying if they can, and the fear, anxiety, and avoidance cause significant distress and impair their ability to function. Take-off, bad weather, and turbulence appear to be the most anxiety provoking aspects of flying. The most extreme manifestations can include panic attacks or vomiting at the mere sight or mention of an aircraft or air travel. Around 60% of people with fear of flying report having some other anxiety disorder. The causes of flight phobia and the mechanisms by which it is maintained were
Proof that π is irrational sine and cosine function are given by sin' = cos and cos' = −sin. Hence the product rule implies By the fundamental theorem of calculus Since and (here we use the above-mentioned characterization of as a zero of the sine function), Claim 2 follows. Conclusion: Since and for (because is the "smallest" positive zero of the sine function), Claims 1 and 2 show that is a "positive" integer. Since and for , we have, by the original definition of "f", which is smaller than 1 for large "n", hence for these "n", by Claim 2. This is impossible for the