Given that, #7cosx+24sinx=Rcos(x-theta), R gt 0, theta" acute"#.
We expanding #cos(x-theta)#, & get,
#7cosx+24sinx=Rcosxcostheta+Rsinxsintheta#.
#"Comparing the respective co-efficients of "cosx and sinx,"#
we have, #Rcostheta=7 and Rsintheta=24#.
Squaring & adding , #R^2(cos^2theta+sin^2theta)=7^2+24^2, #
# or, R^2=25^2," giving, "R=+25...[because, R gt 0]#.
Now, #R=25rArrcostheta=7/R=7/25, &," similarly, "sintheta=24/25#.
Alternatively, #tantheta=24/7. :. theta=arctan(24/7)#.
Altogether, we have,
#7cosx+24sinx=25cos(x-theta), theta=arctan(24/7)#.
Knowing that, #-1 le cos(x-theta) le 1,# we, on multiplication
by #25 gt 0#, have,
# -25 le 25cos(x-theta) le 25, #
# i,e., -25 le 7cosx+24sinx le 25#.
Adding #-12, -37 le 7cosx+24sinx-12 le 13#.
Clearly, #min.{7cosx+24sinx-12}=-37, and, #
#max.{7cosx+24sinx-12}=13#.
