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Example

Example. Express 3cosx° + 4sinx° in the form kcos(x -  )°. *********. a = 3 & b = 4. kcos  °>0 Q1 or Q4. kcos  ° = 3 and ksin  ° = 4. ksin  °>0 Q1 or Q2. (kcos  °) 2 + (ksin  °) 2 = 3 2 + 4 2. k 2 = 9 + 16. . . . k 2 = 25. . k = 5.  in Q1.

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Example

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  1. Example Express 3cosx° + 4sinx° in the form kcos(x - )° ********* a = 3 & b = 4 kcos°>0 Q1 or Q4 kcos° = 3 and ksin° = 4 ksin°>0 Q1 or Q2 (kcos°)2 + (ksin°)2 = 32 + 42 k2 = 9 + 16    k2 = 25  k = 5  in Q1 tan° = b/a = 4/3 tan-1(4/3) = 53.1° So 3sinx° + 4cosx° = 5cos(x – 53.1)°

  2. Example Express –3sinx° + 4cosx° in the form kcos(x - )° ********* –3sinx° + 4cosx° = 4cosx° – 3sinx° (NB cos first ) a = 4 & b = -3 kcos°>0 Q1 or Q4 kcos° = 4 and ksin° = -3 ksin°<0 Q3 or Q4 (kcos°)2 + (ksin°)2 = 42 + (-3)2 k2 = 16 + 9  k2 = 25    k = 5  in Q4 tan° = b/a = -3/4 tan-1(3/4) = 36.9° Q4:  = 360 – 36.9 = 323.1 So –3sinx° + 4cosx° = 5cos(x – 323.1)°

  3. Example (radians) Express -3cos - sin in the form kcos( - ) ********* a = -3 & b = -1 kcos<0 Q2 or Q3 kcos = -3 and ksin = -1 ksin<0 Q3 or Q4 (kcos)2 + (ksin)2 = (-3)2 + (-1)2 k2 = 3 + 1  k2 = 4    k = 2  in Q3 tan° = b/a = -1/-3 = 1/3 tan-1(1/3) = 30° = /6 Q3:  =  + /6 = 7/6 So -3cos - sin = 2cos( – 7/6)

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