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.