一道数学题求反函数:y=5sin(x+π/3)(-3π/4≤x≤π/4）
要过程
y=5sin(x+π/3)在-3π/4≤x≤π/4上不单调,反函数由两部分组成:
(1) -3π/4≤x≤π/6时, -π/2<-5π/12≤x+π/3≤π/2,
∵ sin(-5π/12)=-(√6+√2)/4, ∴ -5(√6+√2)/4≤y≤5,
∴ x+π/3=arcsin(y/5),x=arcsin(y/5)-π/3,
∴ 反函数为y=arcsin(x/5)-π/3,-5(√6+√2)/4≤x≤5。 。。①。
(2) π/6≤x≤π/4时, -π/2<5π/12≤π-(x+π/3)≤π/2,
∵ sin(5π/12)=(√6+√2)/4, ∴ 5(√6+√2)/4≤y≤5,
而...全部
y=5sin(x+π/3)在-3π/4≤x≤π/4上不单调,反函数由两部分组成:
(1) -3π/4≤x≤π/6时, -π/2<-5π/12≤x+π/3≤π/2,
∵ sin(-5π/12)=-(√6+√2)/4, ∴ -5(√6+√2)/4≤y≤5,
∴ x+π/3=arcsin(y/5),x=arcsin(y/5)-π/3,
∴ 反函数为y=arcsin(x/5)-π/3,-5(√6+√2)/4≤x≤5。
。。①。
(2) π/6≤x≤π/4时, -π/2<5π/12≤π-(x+π/3)≤π/2,
∵ sin(5π/12)=(√6+√2)/4, ∴ 5(√6+√2)/4≤y≤5,
而y=5sin[π-(x+π/3)]=5sin(x+π/3)
∴ 2π/3-x=arcsin(y/5),x=2π/3-arcsin(y/5),
∴ 反函数为y=2π/3-arcsin(x/5),-5(√6+√2)/4≤x≤5。
。。②。
①和②合起来就是所要求的反函数。
。收起
