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
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°
RS = √13 cos(5969) = 1.8196
........ how about that !!
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