#rcos(theta+alpha)=r(costhetacosalpha+sinthetasinalpha)#
#=rcosthetacosalpha+rsinthetasinalpha)#
Comparing this to the equation
#3costheta-4sintheta#
We see that
#rcosalpha=3#,
and
#rsinalpha=-4#
Therefore,
#(rcosalpha)^2+(rsinalpha)^2=r^2#
#r^2=3^2+(-4)^2=9+16=25#
Therefore,
#r=sqrt(25)=5#
#cosalpha=3/5#
and
#sinalpha=-4/5#
#alpha=-53.13^@#, #[mod 2pi]#
The equation is
#3costheta-4sintheta=-1.5#
#5cos(theta-53.13^@)=-1.5#
#cos(theta-53.13^@)=-1.5/5=-0.3#
#theta-53.13^@=107.46^@# and #theta-53.13^@=252.54^@#
#theta=107.46+53.13=160.6^@#
and
#theta=252.54+53.13=305.7^@#
The maximum and minimum values of #(3costheta-4sintheta)# are the max. and min. of #(5cos(theta-53.13^@))# which are
#Max. =5#
#Min. =-5#
graph{3cosx-4sinx [-16.02, 16.01, -8.01, 8.01]}
