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Contents
R-Formulae
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R-Formulae
The
R-formulae are useful for re-expressing expressions which are sums of sines
and cosines, which cannot be done using the factor formulae.
a cosx + b sinx = R cos (x - K)
a cosx - b sinx = R cos (x + K)
a sinx + b cosx = R sin (x + K)
a sinx - b cosx = R sin (x - K)
where
 , 
Derivation:
a cosx + b sinx = R cos (x - K)
= R cosx cosK + R sinx sinK
Equating
coefficients of cosx,
R cos K = a ....(1)
Equating
coefficients of sinx,
R sin K = b ....(2)
(1)2
+ (2)2 : (R
cos K)2 + (R sin K)2 = a2 + b2
R2 (cos2K + sin2K) = a2 + b2
R2 = a2 + b2
(2)
/ (1)
:
tan K = b/a
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