Given that 7 cos x + 24 sin x = R cos (x - theta)7cosx+24sinx=Rcos(xθ) where R > 0 and theta is acute, how do you find the minimum and maximum values of 7 cos x + 24 sin x - 127cosx+24sinx12 and the corresponding values of x?

1 Answer
Aug 4, 2018

min.{7cosx+24sinx-12}=-37, and,

max.{7cosx+24sinx-12}=13.

Explanation:

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.