For students and parents · O-Level A-Math
The R-Formula, Shown Working: a sin x + b cos x in One Move
Rewriting a sin x + b cos x as a single wave R sin(x + alpha).
Mrs Eileen Toh
Founder & Curriculum Architect · ex-MOE · 3 min read · Updated 29 Jun 2026
Two trig terms added together, a sin x + b cos x, look as if they cannot be solved or simplified. The R-formula is the move that rewrites them as a single wave, R sin(x + alpha), and the moment a student sees that one expression instead of two, three of the most-asked A-Math questions open up at once: the maximum value, the minimum value, and the solution of the equation.
It is one of the highest-return structures in the syllabus, because the work to learn it is small and the number of questions it unlocks is large.
What R and alpha are
For a sin x + b cos x written as R sin(x + alpha) with a and b positive:
R = √(a2 + b2), and R is always positive.
alpha is the acute angle with tan(alpha) = b / a.
That is the whole formula. R is the height of the combined wave; alpha is how far the wave is shifted.
Worked example, the full question
Express 3 sin x + 4 cos x in the form R sin(x + alpha), then find its maximum value and solve 3 sin x + 4 cos x = 2.5 for 0 <= x <= 360 degrees.
First R and alpha:
R = √(32 + 42) = √(9 + 16) = √(25) = 5.
tan(alpha) = 4/3, so alpha = 53.13 degrees to 2 decimal places.
So 3 sin x + 4 cos x = 5 sin(x + 53.13).
The maximum. A sine wave never goes above 1, so the largest the whole expression can be is R, which is 5. It happens when sin(x + 53.13) = 1, that is when x + 53.13 = 90, giving x = 36.87 degrees. The minimum, by the same reasoning, is minus 5.
The equation. Replace the two terms with the single wave: 5 sin(x + 53.13) = 2.5, so sin(x + 53.13) = 0.5. The angle whose sine is 0.5 is 30 degrees, and sine is also positive at 150 degrees, so within range x + 53.13 = 150 or 390. Subtracting 53.13 gives x = 96.87 or x = 336.87 degrees.
Substituting both answers back into 3 sin x + 4 cos x returns 2.5 in each case, and the maximum of 5 sits at x = 36.87. The answer holds.
A second one, with a minus sign
Express 5 sin x - 12 cos x as R sin(x - alpha). The minus sign just sends you to the R sin(x - alpha) form. R = √(52 + 122) = √(169) = 13, and tan(alpha) = 12/5, so alpha = 67.38 degrees. The expression is 13 sin(x - 67.38), its maximum value is 13, and its minimum is minus 13. Same machine, one sign changed.
The trap
The mistake is reaching for the wrong shift angle, or decimalising R when the question wanted it exact. The fix is the discipline of writing the form first, R sin(x + alpha) or R sin(x - alpha), deciding it from the signs of the two terms, and only then computing R and alpha. Once the form is on the page, the maximum is just R, the minimum is minus R, and the equation is one line of work.
This is the same lesson GPA teaches throughout A-Math trigonometry: name the structure before you compute, and a question that looked like three separate problems becomes one. If your child can find R but freezes on the maximum or the equation, that is the join we work on. GPA's Secondary Math programme builds these structure-first reflexes on real A-Math papers, and a short diagnostic consult will pinpoint where the routine breaks.
Every worked value on this page was checked independently before publishing.