How are complex numbers and trig functions related?
- A sinusoidal function of the form a sin(bx + c)
- a represents the amplitude
- b represents the period (also known as frequency)
- c represents the phase shift
- The function may be written a sin(bx + bc) = a sinb(x + c) where the phase shift is represented by bc
- This will be made clear in the exam
- When written in modulus-argument form the imaginary part of a complex number relates only to the sin part and the real part relates to the cos part
- This means that the complex number can be rewritten in Euler's form and relates to the sinusoidal functions as follows:
- a sin(bx + c) = Im (aei(bx + c))
- a cos(bx + c) = Re (aei(bx + c))
- Complex numbers are particularly useful when working with electrical currents or voltages as these follow sinusoidal wave patterns
- AC voltages may be given in the from V = a sin(bt + c) or V = a cos(bt + c)
How are complex numbers used to add two sinusoidal functions?
- Complex numbers can help to add two sinusoidal functions if they have the same frequency but different amplitudes and phase shifts
- e.g. 2sin(3x + 1) can be added to 3sin(3x - 5) but not 2sin(5x + 1)
- To add asin(bx + c) to dsin(bx + e)
- or acos(bx + c) to dcos(bx + e)
- STEP 1: Consider the complex numbers z1 = aei(bx + c) and z2 = dei(bx + e)
- Then asin(bx + c) + dsin(bx + e) = Im (z1 + z2)
- Or acos(bx + c) + dcos(bx + e) = Re (z1 + z2)
- STEP 2: Factorise z1 + z2 = aei(bx + c) + dei(bx + e) = eibx (aeci + deei)
- STEP 3: Convert aeci + deei into a single complex number in exponential form
- You may need to convert it into Cartesian form first, simplify and then convert back into exponential form
- Your GDC will be able to do this quickly
- STEP 4: Simplify the whole term and use the rules of indices to collect the powers
- STEP 5: Convert into polar form and take...
- only the imaginary part for sin
- or only the real part for cos
Two AC voltage sources are connected in a circuit. If and find an expression for the total voltage in the form .