Harmonic Form (Cambridge International AS Mathematics)

R addition formulae Rcos Rsin etc

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:

• To find α, first divide the b expression by the a expression

  • The R's cancel
  • Use the identity sin α / cos α = tan α
  • You get b / a = tan **α **
  • Then use** tan**^-1(b/a) to find α

How do I write trigonometric expressions in harmonic form?

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

Exam Hint

• There is more than one way to rewrite a given expression, and the question in the exam will tell you which form to use.

Worked Example