DP IB Maths: AI HL

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IBMaths: AI HLDPRevision Notes1. Number & Algebra1.6 Further Complex Numbers1.6.3 Applications of Complex Numbers

1.6.3 Applications of Complex Numbers

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Frequency & Phase of Trig Functions

How are complex numbers and trig functions related?

  • A sinusoidal function is 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 form 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

Exam Tip

  • An exam question involving applications of complex numbers will often be made up of various parts which build on each other
    • Remember to look back at your answers from previous question parts to see if they can help you, especially when looking to convert from Euler's form to a sinusoidal graph form  

Worked example

Two AC voltage sources are connected in a circuit.  If V subscript 1 equals 20 sin invisible function application open parentheses 30 t close parenthesesand V subscript 2 equals 30 sin left parenthesis 30 t plus 5 right parenthesis find an expression for the total voltage in the form V equals A sin invisible function application open parentheses 30 t plus B close parentheses.                      

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