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Higher Maths · Relationships and Calculus

Wave Function | Higher Maths

The Wave Function lets you rewrite expressions of the form k sin x + a cos x in the simpler R cos(x − α) form. This makes it easy to find max/min values and solve equations. It is one of the highest-mark Paper 2 topics — often 7–9 marks.

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SQA Higher MathsSpecification: Relationships and CalculusUnit 3 (legacy)

Wave function form

Express k sin x + a cos x as R cos(x − α) or R sin(x ± α).
R = √(k2 + a2)
α is determined by the signs of the coefficients (use a CAST diagram).
Maximum value of R cos(x − α) is R; minimum is −R.

Worked example

Worked example — Convert and solve

Problem: Express √3 sin x + cos x in the form R cos(x − α) with R > 0 and 0 < α < 2π, and use it to solve √3 sin x + cos x = 1 for 0 ≤ x ≤ 2π.

  1. Expand R cos(x − α) using the compound angle formula.
    R cos(x − α) = R cos x cos α + R sin x sin α
  2. Match coefficients.
    R sin α = √3 (sin coefficient); R cos α = 1 (cos coefficient)
  3. Find R.
    R2 = (√3)2 + 12 = 4 ⇒ R = 2
  4. Find α using tan α = sin/cos coefficients.
    tan α = √3 / 1 ⇒ α = π/3 (1st quadrant since both positive). So √3 sin x + cos x = 2 cos(x − π/3).
  5. Solve 2 cos(x − π/3) = 1.
    cos(x − π/3) = 1/2 ⇒ x − π/3 = π/3 or 5π/3
  6. Solve for x.
    x = 2π/3 or 2π

Practice questions

Try these SQA-style questions. Tap "Show answer" to check your working.

Practice questions

  1. Express 4 sin x + 3 cos x in the form R sin(x + α).
    Show answer
    R = 5, tan α = 3/4 ⇒ α ≈ 0.6435 rad. 5 sin(x + 0.6435)
  2. Find the maximum value of 4 sin x + 3 cos x and the value of x at which it occurs.
    Show answer
    Maximum = 5, when x + α = π/2 ⇒ x ≈ 0.927 rad
  3. Express 3 cos x − 4 sin x in the form R cos(x + α).
    Show answer
    R = 5, R cos α = 3, R sin α = 4 ⇒ tan α = 4/3 ⇒ α ≈ 0.927. 5 cos(x + 0.927)
  4. Solve 4 sin x + 3 cos x = 2.5 for 0 ≤ x ≤ 2π.
    Show answer
    5 sin(x + 0.6435) = 2.5 ⇒ sin(x + 0.6435) = 0.5 ⇒ x + 0.6435 = π/6 or 5π/6 (then add 2π). Solving: x ≈ 1.88 or 5.93
  5. What is the minimum value of 7 cos x − 24 sin x?
    Show answer
    R = √(49 + 576) = 25. Minimum = −25

Common mistakes

Common mistakes & how to avoid them

  • Choosing the wrong quadrant for α — the quadrant depends on the signs of the original coefficients.
  • Setting up R sin α and R cos α the wrong way round (e.g. expanding as cos but matching coefficients as if it were sin).
  • Forgetting to convert the compound angle expansion exactly: cos(x − α) = cos x cos α + sin x sin α (note the +).

Frequently asked questions

Why convert to wave function form?
Once written as R cos(x − α), it is straightforward to find maxima, minima and to solve equations because there is only one trig term.
Is R always positive?
In standard SQA convention, yes — R > 0 is taken as the natural choice. The sign is absorbed into the value of α.
Which form should I use: R sin or R cos?
The question usually specifies. If not, choose whichever expansion most cleanly matches the coefficients of your expression.

Related Higher Maths topics

These topics often appear together in SQA exam questions.

Compound Angle FormulaeTrigonometric FunctionsFunctions & Graphs

← All Higher Maths topics

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