7.3PureStretch

The Double-Angle Formulae

Setting B = A in the addition formulae gives the double-angle formulae. The cosine version has three equivalent forms — choose the one that suits the problem.

28 min Video by Zeeshan Zamurred Trigonometry and Modelling

Edexcel A level Maths: 7.3 The Double Angle FormulaeWatch the full walkthrough before the notes below.

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What you'll be able to do

Derive the double-angle formulae
Know the three forms of cos 2A
Use them to simplify and solve
Apply them to halve powers of sin/cos

The formulae

Putting $B = A$ into the addition formulae:

sin 2 A = 2 sin A cos A

Sine double angle.

cos 2 A = cos^{2} A - sin^{2} A = 2 cos^{2} A - 1 = 1 - 2 sin^{2} A

Three forms of cosine.

tan 2 A = \frac{2 tan A}{1 - tan ^{2} A}

Tangent double angle.

Choosing the cos form

The forms $2 cos^{2} A - 1$ and $1 - 2 sin^{2} A$ are ideal for rewriting $cos^{2} A$ or $sin^{2} A$ — useful in integration too.

1From

cos 2 A = 2 cos^{2} A - 1

2Rearrange:

cos^{2} A = \frac{1}{2} (1 + cos 2 A)

Answer

cos^{2} A = \frac{1}{2} (1 + cos 2 A)

Tip — Pick the cos 2A form whose “leftover” function matches what you want to keep.

Formula recap

sin 2 A = 2 sin A cos A

Sine.

cos 2 A = 2 cos^{2} A - 1 = 1 - 2 sin^{2} A

Cosine (two power-reducing forms).

tan 2 A = \frac{2 t a n A}{1 - t a n ^{2} A}

Tangent.

Common mistakes to avoid

Writing sin 2A = 2 sin A.

sin 2A = 2 sin A cos A.

Forgetting cos 2A has three forms.

cos²−sin², 2cos²−1, and 1−2sin² are all valid.

Key takeaways

sin 2A = 2 sinA cosA.
cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A.
tan 2A = 2tanA/(1 − tan²A).

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