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Published bySeamus Killingbeck Modified about 1 year ago

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

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

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VARIATIONS acosx° + bsinx° = kcos(x - )° asinx° + bcosx° = ksin(x + )° The usual formats are The variations kcos(x + )° and ksin(x - )° are easily obtained by considering reverse rotations from the X-axis. NB:-200° = +160° while +120° = -240° Sosin(x + 225)° = sin(x – 135)° and cos( - 7 / 5 ) = cos( + 3 / 5 )

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

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