F.4 A.Maths

If the equation 4cos(x+45°)+ksin(x+45°)=psinx holds for all values of x, find the values of the constants k and p.

As 4cos(x+45°) + ksin(x+45°) = psinx ... (#)

holds for all real values of x

Put x = 0° into (#)

4cos(0°+45°) + ksin(0°+45°) = psin(0°)

4cos45° + ksin45° = p(0)

4(√2/2) + k(√2/2) = 0

4 + k = 0

k = -4

Put x = -45° into (#)

4cos(-45°+45°) + ksin(-45°+45°) = psin(-45°)

4cos0° + ksin0° = p(-sin45°)

4(1) + k(0) = -p(√2/2)

p = -8/√2

p = -4√2