First off, we need to express these function in the form #y= acos(x+A)+b#
In this way, we can easily deduce the minimum and maximum values, by using the max and min values of #cosx#,
Which are : #1# and #-1# respectively.
#color(red)((a))#
#y=3sinx+4cosx+2#
First express #3sinx+4cosx" "# as #" "acos(x+B)#
So, you want to look for the #a, A# and #B#
#=>3sinx+4cosx=acos(x+A)#
#=>3sinx+4cosx=acosxcosA-asinxsinA#
Equate the coefficients of #sinx# and #cosx#
#=>acosA=4#
#-asinA=3=>asinA=-3#
#(asinA)/(acosA)=(-3)/4=>tanA=-3/4=>A~=-36.87º#
and
#(acosA)^2+(asinA)^2=(4)^2+(-3)^2#
#=>a^2=25=>a=5#
Hence,
#=>3sinx+4cosx=5cos(x-36.87º)#
Thus, #y=3sinx+4cosx+2=5cos(x-36.87º)+2#
The maximum value of #y# occurs when #cosx=1#
#y_"max"=5(1)+2=color(blue)(7)#
The minimum value of #y# occurs when #cosx=-1#
#y_"min"=5(-1)+2=color(blue)(-3)#
#color(red)((b))#
#y=6cosx-4sinx-2#
Similar steps as in #(a)# are repeated,
First express#6cosx-4sinx" "# as #" "acos(x+B)#
So, you want to look for the #a, A# and #B#
#=>6cosx-4sinx=acos(x+A)#
#=>6cosx-4sinx=acosxcosA-asinxsinA#
Equate the coefficients of #sinx# and #cosx#
#=>acosA=6#
#-asinA=-4=>asinA=4#
#(asinA)/(acosA)=4/6=>tanA=2/3=>A~=33.69º#
and
#(acosA)^2+(asinA)^2=(6)^2+(4)^2#
#=>a^2=52=>a=sqrt(52)#
Hence,
#=>6cosx-4sinx=sqrt(52)cos(x+33.69º)#
Thus, #y=6cosx-4sinx-2=sqrt(52)cos(x+33.69º)-2#
The maximum value of #y# occurs when #cos(x+33.69º)=1#
#y_"max"=sqrt(52)(1)-2=color(blue)(sqrt(52)-2)#
The minimum value of #y# occurs when #cos(x+33.69)=-1#
#y_"min"=sqrt(52)(-1)-2=color(blue)(-sqrt(52)-2)#
