The R-formula is one of the most predictable question types in O-Level Additional Mathematics. It appears in Paper 1 or Paper 2 almost every year, it is worth around 6 to 9 marks, and the method barely changes from paper to paper. If you drill the structure properly, this becomes free marks.
Here is exactly how our tutors approach it.
What the R-formula actually does
The R-formula rewrites an expression like $a\sin\theta + b\cos\theta$, which has two trigonometric terms, as a single trigonometric term:
$$a\sin\theta + b\cos\theta = R\sin(\theta + \alpha)$$
where $R = \sqrt{a^2 + b^2}$ and $\alpha$ is an acute angle with $\tan\alpha = \dfrac{b}{a}$.
Why bother? Because one trigonometric term is easy to solve and easy to maximise. Two terms are not.
The four forms you must recognise
The MOE syllabus expects you to handle all four:
- $a\sin\theta + b\cos\theta = R\sin(\theta + \alpha)$
- $a\sin\theta - b\cos\theta = R\sin(\theta - \alpha)$
- $a\cos\theta + b\sin\theta = R\cos(\theta - \alpha)$
- $a\cos\theta - b\sin\theta = R\cos(\theta + \alpha)$
A reliable rule: match the leading function (sine or cosine), and the sign inside the bracket follows the sign in the question for the sine forms, and flips for the cosine forms. In every case $R = \sqrt{a^2+b^2}$ and $\tan\alpha = \dfrac{b}{a}$ where $a$ is the coefficient of the leading function.
If you are unsure, expand your answer using the addition formula and compare coefficients. That takes 30 seconds and guarantees you are right.
Worked example: solving an equation
Express $3\sin x + 4\cos x$ in the form $R\sin(x+\alpha)$, and hence solve $3\sin x + 4\cos x = 2$ for $0^\circ \le x \le 360^\circ$.
Step 1: Find R. $R = \sqrt{3^2 + 4^2} = 5$
Step 2: Find α. $\tan\alpha = \dfrac{4}{3}$, so $\alpha = 53.13^\circ$ (store the full value in your calculator).
So $3\sin x + 4\cos x = 5\sin(x + 53.13^\circ)$.
Step 3: Substitute and isolate. $5\sin(x + 53.13^\circ) = 2$, so $\sin(x + 53.13^\circ) = 0.4$.
Step 4: Shift the range. This is the step most students skip. Since $0^\circ \le x \le 360^\circ$, then $53.13^\circ \le x + 53.13^\circ \le 413.13^\circ$.
Step 5: Basic angle and quadrants. Basic angle $= \sin^{-1}(0.4) = 23.58^\circ$. Sine is positive in quadrants 1 and 2, so within the shifted range: $x + 53.13^\circ = 156.42^\circ$ or $383.58^\circ$ (the second is $360^\circ + 23.58^\circ$).
Step 6: Subtract α. $x = 103.3^\circ$ or $x = 330.4^\circ$ (to 1 decimal place).
Notice that $23.58^\circ$ itself is rejected because it lies outside the shifted range. Writing the shifted range down explicitly is what stops you losing that mark.
Maximum and minimum value questions
Once the expression is $5\sin(x + 53.13^\circ)$, the rest is almost trivial because $\sin$ ranges from $-1$ to $1$.
- Maximum value $= 5$, occurring when $\sin(x+53.13^\circ) = 1$, so $x + 53.13^\circ = 90^\circ$, giving $x = 36.9^\circ$.
- Minimum value $= -5$, occurring when $x + 53.13^\circ = 270^\circ$, giving $x = 216.9^\circ$.
Examiners love variations. Watch for these:
- Added constant: the maximum of $3\sin x + 4\cos x + 7$ is $5 + 7 = 12$.
- Reciprocal: for $\dfrac{10}{3\sin x + 4\cos x + 8}$, the denominator ranges from $3$ to $13$, so the expression ranges from $\dfrac{10}{13}$ to $\dfrac{10}{3}$. The maximum of the fraction comes from the minimum of the denominator.
- Squared or modulus: always sketch or reason about the inner range first.
A full-mark answer states both the value and the angle at which it occurs. Half the marks in these parts are for the angle.
Mistakes that quietly cost marks
- Calculator in radians. If the question uses $0 \le x \le 2\pi$, switch to radians and give $\alpha$ in radians too.
- Rounding α too early. Use the stored value. Rounding $53.13^\circ$ to $53^\circ$ can shift your final answer by a whole degree.
- Forgetting to shift the range. This produces missing or extra solutions, and it is the single most common error we see in marking.
- Taking α as obtuse. By definition $\alpha$ is acute, so $\tan\alpha = \left|\dfrac{b}{a}\right|$ and the sign is handled by the form you choose.
- Not answering in the required accuracy. Degrees to 1 decimal place, radians to 3 significant figures, unless told otherwise.
How to practise this efficiently
Do not do 40 random questions. Do six, one of each type: sine form, cosine form, equation solving in degrees, equation solving in radians, maximum and minimum with a constant, and a reciprocal question. Time yourself at eight minutes each. Once all six are clean, this topic is banked.
If you would like a tutor who sat these papers recently to walk you through a full R-formula question line by line, message us on WhatsApp to book a free trial class: start here.