Applications of Complex Numbers (DP IB Applications & Interpretation (AI)): Revision Note

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 (ae^{i(bx + c)})

  • a cos(bx + c) = Re (ae^{i(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 = ae^{i(bx + c)} and z₂ = de^{i(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  = ae^{i(bx + c)} + de^{i(bx + e)} = e^ibx (ae^ci + de^ei)

• STEP 3: Convert ae^ci + de^ei  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