Find the range of:
#f(x)=3sin(x) +8cos(x-pi/3)+5#
I shall find the range of:
#y = 3sin(x) +8cos(x-pi/3)#
and then add 5.
Substitute #cos(x-pi/3) = cos(pi/3)cos(x) + sin(pi/3)sin(x)#
#y = 3sin(x) +8(cos(pi/3)cos(x) + sin(pi/3)sin(x))#
#y = 3sin(x) + 4cos(x)+4sqrt3sin(x)#
#y = (4sqrt3+3)sin(x) + 4cos(x)" [1]"#
Use the identity for the sine of the sum of two angles:
#y = Asin(x+C) = Acos(C)sin(x)+Asin(C)cos(x)" [2]"#
Matching factors of equation [1] with equation [2], we obtain two equations:
#Acos(C) = 4sqrt3+3" [3]"#
#Asin(C) = 4" [4]"#
Divide equation [4] by equation [3]:
#(Asin(C))/(Acos(C)) = 4/(4sqrt3+3)#
#tan(C) = 4/(4sqrt3+3)#
#C = tan^-1(4/(4sqrt3+3))#
We can use equation [4] to find the value of A:
#Asin(tan^-1(4/(4sqrt3+3))) = 4#
Use the identity #sin(tan^-1(4/(4sqrt3+3))) = (4/(4sqrt3+3))/sqrt(1+(4/(4sqrt3+3))^2)#
#A(4/(4sqrt3+3))/sqrt(1+(4/(4sqrt3+3))^2) = 4#
#A1/sqrt((4sqrt3+3)^2+16) = 1#
#A= sqrt((4sqrt3+3)^2+16)#
#A= sqrt(48+24sqrt3+9+16)#
#A = sqrt(73+24sqrt3)#
Therefore, an alternate form for the original function is:
#y = sqrt(73+24sqrt3)sin(x+tan^-1(4/(4sqrt3+3)))+5#
We know that the sine function varies between -1 and 1, therefore, the range of this function is:
#5-sqrt(73+24sqrt3)<= y <= 5+sqrt(73+24sqrt3), y in RR#
