Question

证明对于任意实数t,复数z=+i的模r,适合不等式r≤.

Answer 1

证明:r=,为证对于任意实数t有r≤,

只要证|cost|+|sint|≤即可.

(1)当kπ≤t≤kπ+(k∈Z)时,则sint·cost≥0,依推论1,|cost|+|sint|=|sint+cost|=|sin(t+)|≤

(2)当kπ+＜t＜(k+1)π(k∈Z)时,sint·cost＜0,sint·(-cost)＞0,依推论1,|cost|+|sint|=|-cost|+|sint|=|sint-cost|=|sin(t-)|≤.

总之,对于任意实数t,有|cost|+|sint|≤成立,即有r≤成立.