Question

Find the range of 7 cos x - 24 sin x + 5

Answer 1

we know, if any function , $f(x)=\pm asinx\pm bcosx$ is given ,
then, range of function will be $[-\sqrt{a^2+b^2},\sqrt{a^2+b^2}]$


here, function is 7cosx - 24sinx + 5

first of all, find range of 7cosx - 24 sinx

a = -24 and b = 7

then, range of (7cosx - 24sinx) is $[-\sqrt{(-24)^2+7^2},\sqrt{24^2+7^2}]$ or, $[-25,25]$

e.g., - 25 ≤ 7cosx - 24 sinx ≤ 25

or, -25 + 5 ≤ 7cosx - 24sinx + 5 ≤ 25 + 5

or, - 20 ≤ 7cosx - 24sinx + 5 ≤ 30

hence, range of (7cosx - 24sinx + 5) is [ -20, 30]