Equating Real & Imaginary Parts (Edexcel International AS Further Maths)

Equating Real & Imaginary Parts

How do I equate real and imaginary parts?

• If two complex expressions are equal, then

  • their real parts are equal

  • and their imaginary parts are equal

• If where and are real

  • then (equating real parts)

  • and (equating imaginary parts)

How do I solve equations using real and imaginary parts?

• Introduce for expressions in

• For example, solve

  • Let

  • Substitute in

    • 

  • Expand and collect terms

    • 

  • Equate real and imaginary parts

    • gives

    • gives

  • Substitute back into

    • 

How do I use real and imaginary parts with modulus signs?

• If then

• Helpful modulus rules are:

  • 

  • 

  • 

  • 

    • Proved using and

• For example, find if , and

  • Use

    • 

  • Use

    • so

  • Square both sides and simplify

    • 

    •