Question

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

Answer 1

7 cos x - 24 sin x + 5

√a² +b²  and minimum value - √a² +b² 
49 + 576 = 625
√675 = 25

minimum value of - 25
and maximum value is 30

so, range is -20 and 30.

hope it's help u

Answer 2

we have to find the range of 7cosx - 24sinx + 5

let f(x) = 7cosx - 24sinx + 5

= 25[7/25 cosx - 24/25 sinx ] + 5

let cosα = 7/25 then sinα = 24/25

= 25[cosα cosx - sinα sinx ] + 5

= 25cos(x + α) + 5

we know, range of cosine function is -1 to 1.

so, -1 ≤ cos(x + α) ≤ 1

⇒-25 ≤ 25cos(x + α) ≤ 25

⇒-25 + 5 ≤ 25cos(x + α) + 5 ≤ 25 + 5

⇒-20 ≤ f(x) ≤ 30

⇒-20 ≤ 7cosx - 24sinx + 5 ≤ 30

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