Express x = 2sin(0.5t + 3) + 3cos(0.5t + 1) in the form x = Rsin(0.5t + ø)
All working must be done in radians.
I am stuck trying to solve this question.
I understand that:
asinx + bcosx can be written as Rsin(x + ø)
Where: Tanø = b/a and: R = sqrt(a² + b²)
The values for x (the frequency) in the question however have different numbers added to them. Does anyone know how to get around this? Thanks
Hey:
So here's my attempt. Can someone please check whether ive gone completely wrong with this:
asinx + bcosx can be written as X = Rsin(x + ø)
Where: Tanø = b/a and: R = sqrt(a² + b²)
Expansion of terms =
2sin(0.5t + 3) = 2cos3sin0.5t + 2sin3cos0.5t
3cos(0.5t + 1) = 3cos1cos0.5t - 3sin1sin0.5t
Addition of terms =
2sin(0.5t + 3) + 3cos(0.5t + 1) = (2cos3-3sin1)sin0.5t + (2sin3+3cos1)cos0.5t
Tanø = b/a
Tanø = 3/2
Tanø = 1.5
ø = Tan^-1(1.5)
ø = 0.983
R = sqrt(a^2 + b^2)
R = sqrt(4+9)
R = sqrt (13)
R = 3.606
Therefore x = 3.606sin(0.5t + 1.5)
Is this right or just completely wrong?? lol
Addition of terms =
2sin(0.5t + 3) + 3cos(0.5t + 1) = (2cos3-3sin1)sin0.5t + (2sin3+3cos1)cos0.5t
Tanø = b/a
Tanø = (2sin3+3cos1) / (2cos3-3sin1)
R = sqrt(a^2 + b^2)
R = sqrt((2cos3-3sin1)²+(2sin3+3cos1)²)
R = sqrt (13 - 12 cos3 sin1 + 12 sin3 cos 1)
R = sqrt (13 + 12 sin2)