Harmonic form

Harmonic form is a way of writing an expression a sin x + b cos x as a single sine or cosine wave: R sin(x ± α) or R cos(x ± α). Here R = √(a² + b²) is the amplitude, and the angle α comes from tan α = b ÷ a (or a ÷ b, depending on the form). It turns a hard expression into one you can read at a glance, so you can find its maximum and minimum and solve equations with a single inverse trig step.

The four harmonic forms

AQA A-level Maths asks you to understand and use expressions for a cos θ + b sin θ in the equivalent forms r cos(θ ± α) or r sin(θ ± α) (statement E6). Each form comes from a compound angle formula:

Pick the form whose signs match your expression. For a sin x + b cos x with both terms positive, R sin(x + α) or R cos(x − α) both work; questions usually tell you which to use. Take R > 0 and α acute.

How to write a sin x + b cos x in harmonic form

1. Expand the target form with the compound angle formula.
2. Match the coefficients of sin x and of cos x, giving R cos α and R sin α.
3. Square and add to find R: R² = a² + b², because cos²α + sin²α = 1.
4. Divide to find α: tan α = (R sin α) ÷ (R cos α).
5. Write the answer, with α in degrees or radians as the question asks.

Example 1: √3 sin x + cos x as R sin(x + α)

Matching: R cos α = √3 and R sin α = 1. Then:

So √3 sin x + cos x = 2 sin(x + 30°). A right-angled triangle shows R and α at once: the two matched values are the shorter sides, R is the hypotenuse, and α is the angle next to the R cos α side.

Here x is in radians and y = √3 sin x + cos x. The curve is a sine wave with amplitude 2, shifted left by π/6 (30°). Its maximum of 2 is at x = π/3 and its minimum of −2 at x = 4π/3. Plot both forms in the Custom Graph tool and they lie exactly on top of each other.

Example 2: 7 cos θ − 24 sin θ as R cos(θ + α)

Matching: R cos α = 7 and R sin α = 24. So R = √(49 + 576) = √625 = 25, and tan α = 24 ÷ 7, giving α = 73.7° (1 d.p.). The answer is 25 cos(θ + 73.7°). Choosing the form with a minus sign meant R sin α came out positive, so α is acute.

Harmonic form tells you a wave's amplitude and how far it is shifted. Practise matching waves to their equations in this question from the Trigonometry course.

Maximum and minimum values

Since sine and cosine always lie between −1 and 1, R sin(x ± α) and R cos(x ± α) lie between −R and R. That makes maximum and minimum questions quick.

Example 3: the maximum of 2 sin x − 5 cos x

Write it as R sin(x − α): R cos α = 2 and R sin α = 5, so R = √29 = 5.385 and α = tan⁻¹(5 ÷ 2) = 68.2°.

• Maximum value: √29 = 5.39 (3 s.f.), when x − 68.2° = 90°, so x = 158.2°.
• Minimum value: −√29, when x − 68.2° = 270°, so x = 338.2°.

Example 4: a fraction

Find the greatest value of 10 ÷ (7 + 2 sin x − 5 cos x). The fraction is largest when the bottom is smallest. The bottom is 7 + √29 sin(x − 68.2°), which is smallest when the sine is −1: 7 − √29 = 1.615. So the greatest value is 10 ÷ 1.615 = 6.19 (3 s.f.). The bottom is never zero, since 7 is bigger than √29.

Solving equations using harmonic form

Example 5: √3 sin x + cos x = 1 for 0° ≤ x < 360°

The range for x + 30° is 30° to 390°. Sine is ½ at 30°, 150° and 390°, but 390° would give x = 360°, which is outside the range. So x + 30° = 30° or 150°, giving x = 0° or x = 120°. Check: √3 sin 120° + cos 120° = 1.5 − 0.5 = 1.

Check your understanding.

Common mistakes

• Adding the coefficients for the maximum. The maximum of a sin x + b cos x is √(a² + b²), not a + b.
• Getting tan α upside down. tan α = (R sin α) ÷ (R cos α): divide the value you matched with sin α by the one you matched with cos α.
• Choosing a form with the wrong signs, which makes α negative or obtuse. Match the signs first.
• Mixing degrees and radians. If x is in radians, give α in radians too.
• Rounding α too early before solving an equation. Keep it in your calculator.

Harmonic form is built on the compound angle formulae, which have their own guide: compound angle formula. For the shapes of the waves, see trig graphs.

Practise with a short timed paper

Trigonometry comes up across A-level Maths. The daily A-level Maths: Algebra & Functions paper is a short timed paper of 8 questions, with a new one every day.

Read next

Maths · TrigonometryCompound angle formula: sin, cos, tan (A ± B) with examplesThe compound angle formulae for sin(A ± B), cos(A ± B) and tan(A ± B): a proof, exact values, solving equations and the R cos(θ − α) form, with examples.