Double angle formulas: sin 2A = 2 sin A cos A

The double angle formulas are sin 2A = 2 sin A cos A; cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A; tan 2A = 2tan A / (1 − tan²A).

\(\displaystyle \sin2A=2\sin A\cos A\) \(\displaystyle \cos2A=\cos^2A-\sin^2A=2\cos^2A-1=1-2\sin^2A\) \(\displaystyle \tan2A=\frac{2\tan A}{1-\tan^2A}\)

What each letter means

\(A\) any angle
\(2A\) double that angle

When to use it

To solve equations with both 2x and x in them, to integrate sin²x or cos²x, and to find exact values such as sin 2A from sin A.

Worked example

Given \(\cos A=\tfrac13\) and \(A\) is acute, find the exact values of \(\sin2A\) and \(\cos2A\).

\(\sin A=\sqrt{1-\tfrac19}=\tfrac{2\sqrt2}{3}\)
\(\sin2A=2\cdot\tfrac{2\sqrt2}{3}\cdot\tfrac13\), \(\cos2A=2\cdot\tfrac19-1\)

Answer: \(\sin2A=\tfrac{4\sqrt2}{9}\), \(\cos2A=-\tfrac79\)

Common mistake

Writing sin 2A = 2 sin A. Doubling the angle does not double the sine.

On your course

Course	In the exam
Edexcel A Level (9MA0)	Not in the Edexcel booklet: learn it
Edexcel International A Level	Not in the booklet: learn it
AQA or OCR A Level	Check your board’s formula booklet: AQA formulae booklet (PDF) · OCR A Level Maths assessment page

From our own A Level Maths formula sheets, in our words. Official: Pearson's formulae booklet (PDF) · Pearson's IAL formulae booklet (PDF).

Practise and revise

Practise A Level Maths practice
Revise Trigonometry
Revise IAL P3 trig addition formulae
Print Edexcel A Level (9MA0) one-page formula sheet

Double angle formulas formula card: sin 2A = 2 sin A cos A; cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A; tan 2A = 2tan A / (1 − tan²A). A Level Math Revision — Double angle formulas formula card. Download the card (PNG) to save or print it.

Questions

What is the double angle formula?

The double angle formulas are sin 2A = 2 sin A cos A; cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A; tan 2A = 2tan A / (1 − tan²A). A: any angle; 2A: double that angle.

Is the double angle formulas given in the exam?

Edexcel A Level (9MA0): not in the Edexcel booklet: learn it. Edexcel International A Level: not in the booklet: learn it. AQA or OCR A Level: check your board’s formula booklet. This comes from our own A Level Maths formula sheets; your teacher has the official booklet.

Which version of cos 2A should I use?

Choose the one that leaves a single trig function: 1 − 2sin²A if the rest of the equation has sin A, 2cos²A − 1 if it has cos A.

Our own wording, examples and card, checked by A Level Math Revision. Not produced or endorsed by Pearson Edexcel, AQA or OCR.