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7.1PureStretch

The Addition Formulae

The addition (compound angle) formulae express trig functions of a sum or difference of angles. They are the foundation for double-angle formulae, the R-form, and many proofs.

28 min Trigonometry and Modelling
Edexcel A level Maths: 7.1 The Addition FormulaeCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • State the sin, cos and tan addition formulae
  • Use them to find exact values
  • Handle both + and − versions
  • Recognise the sign patterns
1

The formulae

For sine the signs match; for cosine they are opposite; tangent has its own form.

sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B
Sine: signs match.
cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B
Cosine: signs flip.
tan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B\tan(A \pm B) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A\tan B}
Tangent.
2

Finding exact values

Split an awkward angle into a sum or difference of standard angles, e.g. 75°=45°+30°75° = 45° + 30°.

Example 1Find the exact value of sin⁡75°\sin 75°.
1sin⁡(45°+30°)=sin⁡45°cos⁡30°+cos⁡45°sin⁡30°\sin(45° + 30°) = \sin45°\cos30° + \cos45°\sin30°.
2=22⋅32+22⋅12= \tfrac{\sqrt2}{2}\cdot\tfrac{\sqrt3}{2} + \tfrac{\sqrt2}{2}\cdot\tfrac12.
3=6+24= \tfrac{\sqrt6 + \sqrt2}{4}.
Answer
6+24\tfrac{\sqrt6 + \sqrt2}{4}

Tip — For cos, remember the sign flips: cos(A+B) uses a minus.

Formula recap

sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B
Sine addition.
cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B
Cosine addition.
tan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B\tan(A \pm B) = \tfrac{\tan A \pm \tan B}{1 \mp \tan A\tan B}
Tangent addition.

Common mistakes to avoid

Using a plus inside cos(A+B) expansion.
cos(A+B) = cosAcosB − sinAsinB — the sign flips.
Thinking sin(A+B) = sinA + sinB.
It is sinAcosB + cosAsinB.

Key takeaways

  • sin(A±B) = sinAcosB ± cosAsinB.
  • cos(A±B) = cosAcosB ∓ sinAsinB.
  • tan(A±B) = (tanA ± tanB)/(1 ∓ tanAtanB).

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⁡(A+B)\cos(A+B) equals:

    • cos⁡Acos⁡B−sin⁡Asin⁡B\cos A\cos B - \sin A\sin B
    • cos⁡Acos⁡B+sin⁡Asin⁡B\cos A\cos B + \sin A\sin B
    • cos⁡A+cos⁡B\cos A + \cos B
    • sin⁡Acos⁡B+cos⁡Asin⁡B\sin A\cos B + \cos A\sin B
    Show answer and explanation

    Answer: cos⁡Acos⁡B−sin⁡Asin⁡B\cos A\cos B - \sin A\sin B

    Sign flips.

  2. sin⁡(A−B)\sin(A-B) equals:

    • sin⁡Acos⁡B−cos⁡Asin⁡B\sin A\cos B - \cos A\sin B
    • sin⁡Acos⁡B+cos⁡Asin⁡B\sin A\cos B + \cos A\sin B
    • sin⁡A−sin⁡B\sin A - \sin B
    • cos⁡Acos⁡B−sin⁡Asin⁡B\cos A\cos B - \sin A\sin B
    Show answer and explanation

    Answer: sin⁡Acos⁡B−cos⁡Asin⁡B\sin A\cos B - \cos A\sin B

    Signs match.

  3. sin⁡75°\sin 75° as sin⁡(45°+30°)\sin(45°+30°) is:

    • 6+24\tfrac{\sqrt6+\sqrt2}{4}
    • 6−24\tfrac{\sqrt6-\sqrt2}{4}
    • 32\tfrac{\sqrt3}{2}
    • 22\tfrac{\sqrt2}{2}
    Show answer and explanation

    Answer: 6+24\tfrac{\sqrt6+\sqrt2}{4}

    Expand and evaluate.

  4. tan⁡(A+B)\tan(A+B) has denominator:

    • 1−tan⁡Atan⁡B1 - \tan A\tan B
    • 1+tan⁡Atan⁡B1 + \tan A\tan B
    • tan⁡Atan⁡B\tan A\tan B
    • 11
    Show answer and explanation

    Answer: 1−tan⁡Atan⁡B1 - \tan A\tan B

    Opposite sign from numerator.

  5. In sin(A+B), the two terms are joined with a:

    • plus
    • minus
    • times
    • divide
    Show answer and explanation

    Answer: plus

    Signs match for sine.

Next step

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