Harmonic Form (AQA A Level Maths): Revision Note

Harmonic form

What is harmonic form?

• If a and b are positive constants, then

  • a sin x + bcos x can be rewritten as R sin (x + α)

  • a sin x - bcos x can be rewritten as R sin (x - α)

  • a cos x + bsin x can be rewritten as R cos (x - α)

  • a cos x - bsin x can be rewritten as R cos (x + α)

• The forms with R and α are called harmonic forms

  • It is the reverse of expanding the right-hand side out using compound angle formulae

  • It can be seen as factorising

• In all four cases R > 0 and 0 < α < 90°, with:

How do I write trigonometric expressions in harmonic form?

• To rewrite the expression use the appropriate compound angle formula and equate coefficients