Frequency & Phase of Trig Functions (DP IB Applications & Interpretation (AI)): Revision Note

Frequency & phase of trig functions

How are complex numbers related to sinusoidal functions?

• A sinusoidal function is related to an exponential function with a complex argument

• You can rewrite in modulus-argument form

  • 

• The relevant sinusoidal function is the real or imaginary part of the exponential function

  • 

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• For example,

  • 

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• 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 can I add two sinusoidal functions which have the same frequencies?

• STEP 1
  Identify the two related exponential functions

  • e.g. for

    • use and

• STEP 2
  Add the two functions together and factorise

  • e.g.

• STEP 3
  Convert the term in the bracket into a single complex number in Euler's form

  • Use your GDC to do this

    • e.g.

• STEP 4
  Simplify the whole expression and use the rules of indices to collect the powers

  • e.g.

• STEP 5
  Convert into polar form and take

  • only the imaginary part for sin

  • or only the real part for cos

    • e.g.