The R formula (harmonic form) rewrites a cos θ + b sin θ as a single wave, R cos(θ − α) or R sin(θ + α), where R = √(a² + b²) and tan α comes from comparing coefficients. Once the expression is one wave, its maximum (R) and minimum (−R) are obvious, and equations like 3 cos θ + 4 sin θ = 2 become straightforward to solve.
Key results for the R formula
- R cos(θ − α) = R cos θ cos α + R sin θ sin α
- R sin(θ + α) = R sin θ cos α + R cos θ sin α
- R = √(a² + b²), always taken positive
- tan α = (coefficient matched to R sin α) / (coefficient matched to R cos α)
Maximum of R cos(θ − α) is R, when θ = α. Minimum is −R, when θ = α + 180°.
Why can a cos θ + b sin θ be written as one wave?
A cosine wave and a sine wave with the same period always add up to another wave of that period, just with a different height and shifted sideways. That new wave is R cos(θ − α): R is the amplitude and α is the shift.
The point of rewriting it is that a cos θ + b sin θ has two trig terms, so you cannot see its maximum or solve a cos θ + b sin θ = c directly. Written as one term, both problems disappear.
How do you find R and α?
Expand the form the question asks for and compare coefficients. For R cos(θ − α) = R cos θ cos α + R sin θ sin α, matching against a cos θ + b sin θ gives R cos α = a and R sin α = b.
Square and add to get R² = a² + b², so R = √(a² + b²). Divide to get tan α = b / a. Always take R positive, and check α is in the range the question gives.
Which form should you use?
There are four forms, R cos(θ ± α) and R sin(θ ± α), and all are valid. Use the one the question asks for. Expand it, match term by term, and the signs take care of themselves. R = √(a² + b²) every time; only the ratio for tan α depends on the form.
Watch the order in R sin(θ + α): it puts the sin θ term first, so with a sin θ + b cos θ you get R cos α = a and R sin α = b.
Finding the maximum and minimum
A cosine only runs between −1 and 1, so R cos(θ − α) has maximum R when θ − α = 0, and minimum −R when θ − α = 180°. The value is the easy mark; the angle where it happens is usually a separate mark, found by setting the bracket to 0 or 180° and solving.
The trap is an expression like k − R cos(θ − α). With a minus in front, the whole thing is largest when the cosine is most negative, so its maximum occurs at the wave's minimum.
Solving a cos θ + b sin θ = c
Rewrite the left-hand side in R form, divide by R, and you have a transformed-argument equation. Substitute u = θ − α, shift the interval for u, solve, then convert back to θ.
If |c| > R there are no solutions at all, because the wave never reaches that height. Checking that first saves time.
Modelling with the R formula
Tides, temperatures and fairground wheels are modelled by expressions like H = 24 + cos(1.3t) − 4 sin(1.3t). Convert the wave part to R form and the highest value is baseline + R, the lowest is baseline − R, and the time it happens comes from setting the bracket to 0 or π (or 0° or 180°) and solving for t.
Worked examples
Example 1. Express 3 cos θ + 4 sin θ in the form R cos(θ − α), where R > 0 and 0° < α < 90°.
- Expand: R cos(θ − α) = R cos θ cos α + R sin θ sin α.
- Compare coefficients: R cos α = 3 and R sin α = 4.
- Square and add: R² = 9 + 16 = 25, so R = 5.
- Divide: tan α = 4/3, so α = 53.1°, which is in the required range.
Answer: 3 cos θ + 4 sin θ = 5 cos(θ − 53.1°).
Marks: B1 for R = 5, M1 for tan α = 4/3, A1 for α = 53.1°.
Example 2. Using 3 cos θ + 4 sin θ = 5 cos(θ − 53.1°), find the maximum and minimum values of 3 cos θ + 4 sin θ and the values of θ in 0° ≤ θ < 360° where they occur.
- Maximum: the cosine can reach 1, so the greatest value is 5.
- Where: θ − 53.1° = 0, so θ = 53.1°.
- Minimum: the cosine can reach −1, so the least value is −5.
- Where: θ − 53.1° = 180°, so θ = 233.1°.
Answer: Maximum 5 at θ = 53.1°; minimum −5 at θ = 233.1°.
Marks: B1 for each value and each angle.
Example 3. Solve 3 cos θ + 4 sin θ = 2 for 0° ≤ θ < 360°, giving answers to 1 d.p.
- R form, keeping α unrounded: 5 cos(θ − 53.13°) = 2, so cos(θ − 53.13°) = 0.4.
- Let u = θ − 53.13°. The interval becomes −53.13° ≤ u < 306.87°.
- cos⁻¹(0.4) = 66.42°, and cosine is positive, so u = 66.42° or u = −66.42°.
- Convert back: θ = 119.55° or θ = −13.29°, and −13.29° + 360° = 346.71°.
Answer: θ = 119.6°, 346.7° (1 d.p.).
Marks: M1 for using the R form, M1 for shifting the interval, A1 for both answers.
Example 4. Express 8 sin θ + 6 cos θ in the form R sin(θ + α), where R > 0 and 0° < α < 90°.
- Expand: R sin(θ + α) = R sin θ cos α + R cos θ sin α.
- Match the sin θ terms: R cos α = 8. Match the cos θ terms: R sin α = 6.
- R² = 64 + 36 = 100, so R = 10.
- tan α = 6/8 = 0.75, so α = 36.9°.
Answer: 8 sin θ + 6 cos θ = 10 sin(θ + 36.9°).
Marks: B1 for R = 10, M1 for tan α, A1 for α.
Common mistakes
- Taking tan α the wrong way up. Write down R cos α = … and R sin α = … first, then divide sin by cos. Never guess the ratio.
- Using the wrong form. If the question asks for R sin(θ + α), expand that form. A correct answer in a different form scores very little.
- Forgetting to shift the interval. After substituting u = θ − α, the interval for u moves too. Solutions just outside the original interval are easy to miss.
- Giving only the maximum value. Most questions also want the angle or time where it occurs. Set the bracket to 0 or 180° and solve.
- Missing a minus sign in front of the wave. For k − R cos(θ − α), the maximum happens where the cosine equals −1, not 1.
- Rounding α too early. Keep α to at least two decimal places while solving, and round only the final answers. Using α = 53.1° instead of 53.13° can shift a final answer by 0.1°.
How the R formula is examined
In Edexcel International A-Level this sits in Pure 3 (paper WMA13), in the trigonometric addition formulae chapter. In Edexcel A-Level (9MA0) it is Year 2 Pure content. A typical question runs in parts:
- express a cos θ + b sin θ in a stated R form, with R and α to a given accuracy
- write down the maximum or minimum value and where it occurs
- solve an equation using the R form, often in radians
- interpret a model, such as the height of a wheel or the depth of a harbour, using the R form
The Pure 3 addition formulae exam questions include full R formula questions with mark schemes. For extra practice, use the trigonometric formulae worksheet or the trigonometry revision engine.
This builds on the addition formulae, and the further trig equations course shows when to reach for it.
Frequently asked questions
What is R in the R formula?
R is the amplitude of the single combined wave. For a cos θ + b sin θ it is always √(a² + b²), whichever of the four forms you use.
Is the R formula the same as the harmonic form?
Yes. Harmonic form, R formula and auxiliary angle method all describe rewriting a cos θ + b sin θ as one sine or cosine wave.
How do I know whether to use R cos(θ − α) or R sin(θ + α)?
Use whichever the question states. If you get to choose, pick the form whose expansion has the same signs as your expression, so that α comes out positive and acute.
Should α be in degrees or radians?
Match the question. If θ is in degrees, give α in degrees; if the interval is in radians, give α in radians, usually to 3 decimal places.