Harmonic Form (AQA A Level Maths: Pure): Revision Note
Exam code: 7357
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
Examiner Tips and Tricks
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
Unlock more, it's free!
Join the 100,000+ Students that ❤️ Save My Exams
the (exam) results speak for themselves:
Was this revision note helpful?
Build on this topic
Questions students ask about Harmonic Form
Real questions from students revising this topic on Save My Exams
How do you know which harmonic form to use?
Match the leading trig function and the sign between the terms — although in the exam the question nearly always tells you which form to use.
With a and b positive:
- a sin x + b cos x = R sin(x + α)
- a sin x − b cos x = R sin(x − α)
- a cos x + b sin x = R cos(x − α)
- a cos x − b sin x = R cos(x + α)
Quick rule:
- sin form: the sign in the bracket matches the sign in the expression
- cos form: the sign in the bracket is the opposite
Why: cos(x + α) = cos x cos α − sin x sin α, and it is that minus in the expansion that matches a cos x − b sin x. The sin expansion keeps the same sign throughout.
Page examples:
- 5 cos x − 12 sin x: cos first, minus sign → 13 cos(x + 67.38°)
- sin θ − 4 cos θ: sin first, minus sign → √17 sin(θ − 75.96°)
In all four cases R > 0 and 0 < α < 90°.
How do you find the minimum value of an expression in harmonic form?
The minimum value of R sin(x ± α) or R cos(x ± α) is −R, because sine and cosine never go below −1 — and the maximum is R.
- Write the expression in harmonic form to find R
- The minimum value is −R
- To find where it happens, set the sine or cosine part equal to −1 and solve:
- sin(x − α) = −1 when x − α = 270°
- cos(x + α) = −1 when x + α = 180°
Using the page's examples:
- sin θ − 4 cos θ = √17 sin(θ − 75.96°): minimum −√17, at θ − 75.96° = 270°, so θ = 345.96°
- 5 cos x − 12 sin x = 13 cos(x + 67.38°): minimum −13, at x + 67.38° = 180°, so x = 112.62°
Watch out:
- Check your angle lies in the range the question gives; add or subtract 360° if not
- If the harmonic form sits inside something bigger, such as 1 ÷ (20 + 13 cos(x + 67.38°)), the smallest value of the whole thing comes from the largest denominator, so use +R there
How do you derive harmonic form from compound angle formulae?
Expand the harmonic form using the compound angle formula, then equate coefficients with your expression to get R cos α = a and R sin α = b.
Page example: write 5 cos x − 12 sin x as R cos(x + α)
- Expand: R cos(x + α) = R cos α cos x − R sin α sin x
- Compare coefficients with 5 cos x − 12 sin x: R cos α = 5 and R sin α = 12
- Find α by dividing, so the R's cancel: R sin α ÷ R cos α = tan α = 12/5, giving α = tan⁻¹(12/5) = 67.38°
- Find R by squaring and adding: R²cos²α + R²sin²α = 5² + 12², so R²(cos²α + sin²α) = 169. Since cos²α + sin²α = 1, R² = 169 and R = 13
- Write the answer: 5 cos x − 12 sin x = 13 cos(x + 67.38°)
In general: a = R cos α, b = R sin α, so R = √(a² + b²) and tan α = b/a.
Harmonic form is the compound angle formula run backwards, rather like factorising reverses expanding.
