The double angle formulae

The double angle formulae rewrite the sine, cosine or tangent of 2A in terms of the angle A. They are sin 2A = 2 sin A cos A, cos 2A = cos²A − sin²A and tan 2A = 2 tan A / (1 − tan²A). The cos 2A formula has two more forms, 2cos²A − 1 and 1 − 2sin²A, which are the ones you use most for solving equations and integrating.

The double angle formulae

They are identities: true for every value of A, in degrees or radians, wherever both sides are defined. The tan formula fails when tan²A = 1, that is when A = 45°, 135° and so on, because then 2A is 90° or 270° and tan 2A does not exist.

A common trap is to think sin 2A = 2 sin A. Try A = 30°: sin 60° = 0.866, but 2 sin 30° = 1. Doubling the angle does not double the sine.

The three forms of cos 2A

All three come from the first one, using sin²A + cos²A = 1. Replace sin²A with 1 − cos²A to get 2cos²A − 1. Replace cos²A with 1 − sin²A to get 1 − 2sin²A. Choose the form that leaves only one kind of trig function in your equation:

Deriving the double angle formulae

Every double angle formula comes from a compound angle formula with B = A. Start from:

Put B = A, and the two terms on the right are the same:

In the same way, cos(A + B) = cos A cos B − sin A sin B gives cos 2A = cos²A − sin²A. And tan(A + B) = (tan A + tan B)/(1 − tan A tan B) gives tan 2A = 2 tan A/(1 − tan²A).

This matters in the exam, because the compound angle formulae are printed in the formula booklet and the double angle formulae are not. If you forget a double angle formula, you can rebuild it in one line.

Rearranged forms for sin² and cos²

Rearranging the last two forms of cos 2A gives expressions for cos²A and sin²A with no squares, which is what makes them integrable:

Writing A as half an angle gives the half angle versions, such as cos θ = 2cos²(θ/2) − 1 and sin θ = 2 sin(θ/2) cos(θ/2). They are the same formulae with different letters.

The graph shows the first rearrangement. Here x is the angle in radians and y = cos²x.

cos²x is a cosine wave shifted up to sit between 0 and 1, with half the height and twice the frequency of cos x: exactly (1 + cos 2x)/2. That repeating every π, rather than every 2π, is the cos 2x showing through.

Double angle formula examples

Example 1: exact values

Given that cos θ = 5/13 and θ is acute, find the exact values of sin 2θ, cos 2θ and tan 2θ.

First find sin θ. A 5, 12, 13 right-angled triangle gives sin θ = 12/13 (or use sin²θ = 1 − cos²θ = 144/169).

cos 2θ is negative, so 2θ is obtuse, which fits: θ is a bit over 67°, so 2θ is a bit over 134°.

Example 2: solving an equation with sin 2x

Solve sin 2x = sin x for 0° ≤ x ≤ 360°.

Replace sin 2x, then factorise. Do not divide by sin x, or you lose solutions.

sin x = 0 gives x = 0°, 180°, 360°. cos x = ½ gives x = 60°, 300°. So x = 0°, 60°, 180°, 300°, 360°.

Example 3: solving an equation with cos 2x

Solve cos 2x + 3 sin x = 2 for 0° ≤ x ≤ 360°.

The rest of the equation has sin x, so use cos 2x = 1 − 2sin²x. That turns it into a quadratic in sin x.

sin x = ½ gives x = 30° or 150°. sin x = 1 gives x = 90°. So x = 30°, 90°, 150°.

Example 4: proving an identity

Prove that sin 2x / (1 + cos 2x) ≡ tan x.

Work on the left-hand side. The 1 in the denominator suggests cos 2x = 2cos²x − 1, because the 1s cancel.

Example 5: integrating sin²x

Find the integral of sin²x.

There is no direct rule for sin²x, but the rearranged double angle formula turns it into something you can integrate:

This is the standard method at A-level for integrating sin² and cos², and it is why the rearranged forms are worth knowing by heart.

Example 6: a triple angle

Show that cos 3A = 4cos³A − 3cos A.

Write 3A as 2A + A and use the compound angle formula, then the double angle formulae:

Replace sin²A with 1 − cos²A and collect terms: 2cos³A − cos A − 2cos A + 2cos³A = 4cos³A − 3cos A.

Using the double angle formulae in reverse

Exam questions often hide a double angle formula and ask for an exact value without a calculator. Spot the pattern on the right-hand side of a formula, then collapse it into a single trig function of the doubled angle:

• 2 sin 15° cos 15° = sin 30° = ½
• cos²22.5° − sin²22.5° = cos 45° = √2/2
• 1 − 2sin²(π/12) = cos(π/6) = √3/2
• 2 tan 22.5° ÷ (1 − tan²22.5°) = tan 45° = 1

The same trick simplifies expressions: 6 sin x cos x is 3 × 2 sin x cos x, which is 3 sin 2x.

Where sin 2θ turns up in physics

A projectile launched at speed u and angle θ on level ground, ignoring air resistance, lands at a horizontal distance R = u² sin 2θ / g. The double angle formula is what turns 2 sin θ cos θ in the working into sin 2θ. Since sin 2θ is largest when 2θ = 90°, the range is greatest at a launch angle of 45°. You can try it in the Projectile Motion Simulator.

Here is a quick check before the mistakes.

Common mistakes

• Thinking sin 2A = 2 sin A. The 2 is inside the sine. sin 2A = 2 sin A cos A.
• Choosing the wrong form of cos 2A. Pick the form that matches the other trig function in the equation.
• Dividing by sin x or cos x. In sin 2x = sin x, dividing by sin x loses x = 0°, 180° and 360°. Factorise instead.
• Missing solutions in the interval. After finding the principal value, use the graph or CAST diagram to find every solution in the range.
• Halving the interval wrongly. If you solve for 2x directly, double the interval first: 0° ≤ x ≤ 360° means 0° ≤ 2x ≤ 720°.
• Forgetting the ½ in integration. The integral of cos 2x is ½ sin 2x, not sin 2x.

Is the double angle formula in the formula booklet?

No. This was checked against the A-level Maths formulae booklets in November 2026.

So learn the double angle formulae, including all three forms of cos 2A and the rearranged versions for sin²A and cos²A. If you blank, set B = A in the compound angle formulae printed in the booklet.

This practice from the Trigonometry course uses the identities you need alongside the double angle formulae, such as sin²θ + cos²θ = 1. Pick the simplest form, then press Mark to check your answer in the course.

Practise trig identities in a short timed paper

A-level Maths: Algebra and Functions gives you a short timed paper every day, with 8 questions worth 25 to 35 marks, and trigonometric identities and equations come up regularly. In the US, the same practice opens as AP Precalculus. To build up from SOH CAH TOA to identities, the Trigonometry course covers the unit circle, trig graphs and the Pythagorean identity.

Read next

Maths · TrigonometryCompound angle formula: sin, cos, tan (A ± B) with examplesThe compound angle formulae for sin(A ± B), cos(A ± B) and tan(A ± B): a proof, exact values, solving equations and the R cos(θ − α) form, with examples.