express 3 cos x -2 sin x in th eform R cos (x + a) and hence write down the maximum and minimum values of 3 cos x - 2 sin x.

1 answer

let 3cosx - 2sinx = Rcos(x+a)

Rcos(x+a) = R(cosxcosa - sinxsina)
= Rcosxcosa - Rsinxsina

so we have the identity
Rcosxcosa - Rsinxsina = 3cosx-2sinx
this must be valid for any x
so let's pick x's that simplify this

let x = 0
then
Rcos0cosa - Rsin0sins = 3cos0 - 2sin0
Rcosa = 3
cosa = 3/R

let x = 90°
Rcos90cosa - Rsin90sina = 3cos90 - 2sin90
-Rsina = -2
sina = 2/R

but sin^2a + cos^2a = 1
4/R^2 + 9/R^2 = 1
R^2 = 13
R = √13

also : sina/cosa = (2/R) / (3/R) = 23
tana = 2/3
a = arctan (2/3) = 33.69°

thus 3cosx - 2sinx = √13cos(x + 33.69°)

check by taking any angle x
let x = 26°
LS =1.8196...
RS = √13 cos(5969) = 1.8196
........ how about that !!
Similar Questions
  1. Which is not true about the subjunctive mood? (1 point) ResponsesA. It can express something that is feared B. it can express
    1. answers icon 1 answer
  2. Which is not true about the subjunctive mood? (1 point)◎ Il can express something that is hoped for It can express something
    1. answers icon 1 answer
  3. Which is not true about the subjunctive mood? (1 point)A. It can express something that is unlikely B. It can express something
    1. answers icon 1 answer
  4. which is not true about the subjunctive mood?a. it can express something that is hoped for b. it can express something that is
    1. answers icon 1 answer
more similar questions