求下列函数的反函数(需要详细过程)
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求下列函数的反函数(需要详细过程)
(1)y=5sin(x+π/3) (-3π/4≤x≤π/4)
(2)y=1/2arctan(x/3)
(1)y=5sin(x+π/3) (-3π/4≤x≤π/4)
(2)y=1/2arctan(x/3)
-3π/4≤x≤π/4
-5π/12≤x+π/3≤7π/12
x≤π/6时,x+π/3=arcsin(y/5)
x>π/6时,
y=5sin(x+π/3) =5sin(2π/3-x)
2π/3-x=arcsin(y/5)
所以,反函数是:
y≤π/6时,y=arcsin(x/5)-π/3
y>π/6时,y=2π/3-arcsin(y/5)
2)
y=1/2arctan(x/3)
2y=arctanx/3
tan2y=x/3
x=3tan2y
即:反函数为:y=3tan2x
-5π/12≤x+π/3≤7π/12
x≤π/6时,x+π/3=arcsin(y/5)
x>π/6时,
y=5sin(x+π/3) =5sin(2π/3-x)
2π/3-x=arcsin(y/5)
所以,反函数是:
y≤π/6时,y=arcsin(x/5)-π/3
y>π/6时,y=2π/3-arcsin(y/5)
2)
y=1/2arctan(x/3)
2y=arctanx/3
tan2y=x/3
x=3tan2y
即:反函数为:y=3tan2x