1. Example
Express -8sinx° + 15cosx° in the form ksin(x + )°
*********
a = -8 & b = 15
kcos°<0 Q2 or Q3
kcos° = -8 and ksin° = 15
ksin°>0 Q1 or Q2
(kcos°)2 + (ksin°)2 = (-8)
k2 =   
k2 = 289 
k = 17
tan° = b/a
= 15/-8
 in Q2
tan-1(15/8) = 61.9°
Q2:  = 180 – 61.9
= 118.1°
So -8sinx° + 15cosx° = 17sin(x )°

2. Example (radians)
Express 3/2sin - 1/2cos in the form ksin( + )
*********
a = 3/2 & b = -1/2
kcos>0 Q1 or Q4
kcos = 3/2 and ksin = -1/2
ksin<0 Q3 or Q4
(kcos)2 + (ksin)2 = (3/2 )2 + (-1/2)2
k2 = 3/4 + 1/4 
k2 = 1   
k = 1
 in Q4
tan = b/a
= (-1/2)(3/2)
= -1/2 X ( 2 /3)
= -1/3
tan-1 (1/3)= 30° = /6
Q4:  = 2 - /6
= 11/6
So 3/2sin - 1/2cos = sin( + 11/6)

3. VARIATIONS
The usual formats are
acosx° + bsinx° = kcos(x - )°
asinx° + bcosx° = ksin(x + )°
The variations kcos(x + )° and ksin(x - )° are easily obtained by considering reverse rotations from the X-axis.
NB: -200° = +160° while ° = -240°
So sin(x + 225)° = sin(x – 135)°
and cos( - 7/5) = cos( + 3/5)

4. Example
Express 12sinx° - 5cosx° in the form ksin(x - )°
*********
a = 12 & b = -5
kcos° = 12 and ksin° = -5
kcos°>0 Q1 or Q4
(kcos°)2 + (ksin°)2 = (-5)2
ksin°<0 Q3 or Q4
k2 =
k2 = 169 
k = 13   
tan° = b/a
= -5/12
 in Q4
tan-1(5/12) = 22.6°
Q4:  = 360 – 22.6
= 337.4°
So 12sinx° - 5cosx° = 13sin(x )° =
13sin(x – 22.6)°