Back to course
7.2PureStretch

Using the Addition Formulae

The addition formulae are used to prove identities, simplify expressions and solve equations. This lesson focuses on applying them confidently in both directions.

26 min Trigonometry and Modelling
Edexcel A level Maths: 7.2 Using the Addition FormulaeCurated by Kepler, not produced by us. Watch it before the notes below.
Open on YouTube

What you'll be able to do

  • Apply the formulae to prove identities
  • Simplify combined expressions
  • Recognise an expansion in reverse
  • Solve equations using compound angles
1

Reading the formulae in reverse

A combination like sin⁡Acos⁡B+cos⁡Asin⁡B\sin A\cos B + \cos A\sin B collapses to sin⁡(A+B)\sin(A+B). Spotting these reversed forms is half the skill.

sin⁡Acos⁡B+cos⁡Asin⁡B=sin⁡(A+B)\sin A\cos B + \cos A\sin B = \sin(A+B)
Recognise the pattern backwards.
Example 1Simplify cos⁡50°cos⁡10°−sin⁡50°sin⁡10°\cos 50°\cos 10° - \sin 50°\sin 10°.
1Matches cos⁡(A+B)\cos(A+B) with A=50°,B=10°A=50°, B=10°.
2=cos⁡60°=12= \cos 60° = \tfrac12.
Answer
12\tfrac12
2

Proving identities

Expand both compound angles, then simplify using sin⁡2+cos⁡2=1\sin^2 + \cos^2 = 1 and other known identities.

Tip — When an expression has products of sin and cos of two angles, suspect an addition formula in reverse.

Formula recap

cos⁡Acos⁡B−sin⁡Asin⁡B=cos⁡(A+B)\cos A\cos B - \sin A\sin B = \cos(A+B)
Reverse cosine.
sin⁡Acos⁡B+cos⁡Asin⁡B=sin⁡(A+B)\sin A\cos B + \cos A\sin B = \sin(A+B)
Reverse sine.

Common mistakes to avoid

Not recognising a reversed expansion.
Products like cosAcosB − sinAsinB collapse to cos(A+B).
Mismatching the sign when collapsing.
A minus between the products means cosine of a sum.

Key takeaways

  • Use the formulae forwards and backwards.
  • cosAcosB − sinAsinB = cos(A+B); sinAcosB + cosAsinB = sin(A+B).
  • Collapse products to a single compound angle to simplify.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. cos⁡50°cos⁡10°−sin⁡50°sin⁡10°\cos 50°\cos 10° - \sin 50°\sin 10° equals:

    • cos⁡60°\cos 60°
    • cos⁡40°\cos 40°
    • sin⁡60°\sin 60°
    • cos⁡500°\cos 500°
    Show answer and explanation

    Answer: cos⁡60°\cos 60°

    cos(A+B).

  2. sin⁡3θcos⁡θ+cos⁡3θsin⁡θ\sin 3\theta\cos\theta + \cos 3\theta\sin\theta equals:

    • sin⁡4θ\sin 4\theta
    • sin⁡2θ\sin 2\theta
    • cos⁡4θ\cos 4\theta
    • sin⁡3θ\sin 3\theta
    Show answer and explanation

    Answer: sin⁡4θ\sin 4\theta

    sin(3θ+θ).

  3. A minus sign between products of cosines and sines signals:

    • cosine of a sum
    • sine of a sum
    • cosine of a difference
    • tangent
    Show answer and explanation

    Answer: cosine of a sum

    cos(A+B) pattern.

  4. cos⁡60°\cos 60° equals:

    • 12\tfrac12
    • 32\tfrac{\sqrt3}{2}
    • 11
    • 00
    Show answer and explanation

    Answer: 12\tfrac12

    Standard value.

  5. Best clue to use an addition formula in reverse is:

    • products of trig of two angles
    • a single angle
    • a fraction
    • a square root
    Show answer and explanation

    Answer: products of trig of two angles

    That is the expanded form.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

All 7 lessons

Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.