Double angle formulae: sin 2A, cos 2A and tan 2A
The double angle formulae write sin 2A, cos 2A and tan 2A in terms of sin A, cos A and tan A. They come straight from the compound angle formulae and appear throughout A level Maths: in exact values, trigonometric equations, identities and integration.
On this page
The double angle formulae
| Formula | Notes |
|---|---|
| sin 2A = 2 sin A cos A | One form only |
| cos 2A = cos²A − sin²A | The basic form |
| cos 2A = 2cos²A − 1 | Use when you want only cosines |
| cos 2A = 1 − 2sin²A | Use when you want only sines |
| tan 2A = 2 tan A / (1 − tan²A) | Undefined when tan²A = 1 |
Where they come from
Start from the compound angle formulae and put B = A:
- sin(A + B) = sin A cos B + cos A sin B, so sin 2A = sin A cos A + cos A sin A = 2 sin A cos A.
- cos(A + B) = cos A cos B − sin A sin B, so cos 2A = cos²A − sin²A.
- Using sin²A + cos²A = 1 to replace sin²A or cos²A gives the other two forms of cos 2A.
- tan(A + B) = (tan A + tan B)/(1 − tan A tan B), so tan 2A = 2 tan A/(1 − tan²A).
Learn the formulas well enough to write them down at once. Deriving them in the exam works, but it costs time you need for the question.
Rearranged forms you will need
Rearranging the cos 2A formulas gives two results used all the time in integration:
- cos²A = (1 + cos 2A)/2
- sin²A = (1 − cos 2A)/2
Worked examples
Example 1: exact values
Given sin A = 3/5 and A is acute, find the exact values of sin 2A, cos 2A and tan 2A.
- A is acute, so cos A = 4/5 (a 3-4-5 triangle).
- sin 2A = 2 × (3/5) × (4/5) = 24/25.
- cos 2A = 16/25 − 9/25 = 7/25.
- tan 2A = (24/25)/(7/25) = 24/7. Check with tan A = 3/4: 2(3/4)/(1 − 9/16) = 24/7.
Example 2: an equation with sin 2x
Solve sin 2x = sin x for 0° ≤ x ≤ 360°.
- Replace sin 2x: 2 sin x cos x = sin x.
- Bring everything to one side and factorise: sin x (2 cos x − 1) = 0. Never divide by sin x: you would lose solutions.
- sin x = 0 gives x = 0°, 180°, 360°. cos x = 1/2 gives x = 60°, 300°.
- x = 0°, 60°, 180°, 300°, 360°.
Example 3: choosing the right form of cos 2x
Solve cos 2x + 3 sin x = 2 for 0° ≤ x < 360°.
- The equation has sin x, so use cos 2x = 1 − 2 sin²x.
- 1 − 2 sin²x + 3 sin x − 2 = 0, so 2 sin²x − 3 sin x + 1 = 0.
- (2 sin x − 1)(sin x − 1) = 0, so sin x = 1/2 or sin x = 1.
- x = 30°, 90°, 150°.
Example 4: integration
Find ∫ sin²x dx.
- Use sin²x = (1 − cos 2x)/2.
- ∫ (1/2 − (1/2) cos 2x) dx = x/2 − (sin 2x)/4 + C.
Example 5: proving an identity
Prove that (1 − cos 2x)/sin 2x ≡ tan x.
- Top: 1 − cos 2x = 1 − (1 − 2 sin²x) = 2 sin²x.
- Bottom: sin 2x = 2 sin x cos x.
- So the fraction is 2 sin²x / (2 sin x cos x) = sin x / cos x = tan x.
How to choose the form of cos 2A
- The rest of the equation has sin x: use 1 − 2 sin²x, so everything is in sin x.
- The rest has cos x: use 2 cos²x − 1.
- You are integrating sin²x or cos²x: rearrange to the forms in the section above.
- You want to factorise cos²x − sin²x: it is a difference of two squares.
Our trig identities guide lists the other identities you need for A level Maths, and the A level trigonometry guide covers equations and radians.
Practice questions
Try each one before opening the answer.
Q1. Given cos A = 5/13 with A acute, find the exact value of sin 2A.
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Q2. Write cos²x in terms of cos 2x.
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Q3. Solve cos 2x = cos x for 0° ≤ x < 360°.
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Q4. Find the exact value of ∫0π/2 cos²x dx.
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Q5. Find the exact value of sin 15° cos 15°.
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Q6. If tan A = 2, find tan 2A.
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Common mistakes
- Writing sin 2A = 2 sin A: the 2 is inside the function, so this is wrong.
- Dividing both sides by sin x or cos x and losing solutions.
- Choosing a form of cos 2A that leaves both sin and cos in the equation.
- Forgetting the 1/2 when integrating cos 2x: ∫ cos 2x dx = (sin 2x)/2.
Frequently asked questions
What are the double angle formulae?
sin 2A = 2 sin A cos A; cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A; tan 2A = 2 tan A/(1 − tan²A).
Why are there three forms of cos 2A?
They are the same identity, rewritten with sin²A + cos²A = 1, so you can choose the one that suits the question.
Are the double angle formulae in the A level formula booklet?
Check your own board's booklet. Either way, learn them: you will use them so often that looking them up costs time.
What are the half angle formulae?
Rearranging cos 2A gives cos²A = (1 + cos 2A)/2 and sin²A = (1 − cos 2A)/2. Replacing A by A/2 gives half angle versions.
Key takeaways
- sin 2A = 2 sin A cos A, and cos 2A has three forms.
- Choose the form of cos 2A that leaves one trig function in the equation.
- sin²x and cos²x integrate via the cos 2x forms.