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The sin 2x formula is sin 2x = 2 sin x cos x. It is the double angle identity for sine: it gives the sine of twice an angle using the sine and cosine of the angle itself. In terms of tangent, sin 2x = 2 tan x / (1 + tan²x).

Sin 2x formula at a glance

sin 2x = 2 sin x cos x

sin 2x = 2 tan x / (1 + tan²x)

sin²x = (1 − cos 2x) / 2

In this guide:

  • what sin 2x means, with a quick check using 30°
  • every form of the formula, and how each is derived
  • the sin²x formula, and how it differs from sin 2x
  • 6 step-by-step solved examples, plus a separate section for IB students

What is sin 2x?

Sin 2x is the sine of twice the angle x. It is also called the double angle formula of sine.

Quick check with x = 30°:

  • Directly: sin 2x = sin 60° = √3/2
  • With the formula: 2 sin 30° cos 30° = 2 × 1/2 × √3/2 = √3/2 ✓

Notice that 2 sin 30° = 1, which is not √3/2. That is why sin 2x is not the same as 2 sin x.

Domain, range and period of sin 2x

  • Domain: all real numbers
  • Range: −1 ≤ sin 2x ≤ 1
  • Period: π radians (180°), half the period of sin x, because the angle doubles

Image suggestion: sin x and sin 2x on the same axes from 0 to 2π, showing sin 2x completing two cycles.

Sin 2x formula: all forms

1. In terms of sin and cos (the main form)

\sin 2x = 2 \sin x \cos x

2. In terms of tan

\sin 2x = \frac{2 \tan x}{1 + \tan^2 x}

3. In terms of cos only: sin 2x = ±2 cos x √(1 − cos²x)

4. In terms of sin only: sin 2x = ±2 sin x √(1 − sin²x)

Forms 3 and 4 come from sin²x + cos²x = 1. Choose the + or − sign from the quadrant of x.

Derivation of the sin 2x formula

  1. Begin with the compound angle identity for sine: sin(A + B) = sin A cos B + cos A sin B.
  2. Let both angles be x. The left side becomes sin 2x, and the right side becomes sin x cos x + cos x sin x.
  3. The two terms on the right are identical, so they add to give sin 2x = 2 sin x cos x.

Derivation of sin 2x in terms of tan

  1. Take sin 2x = 2 sin x cos x and divide top and bottom by cos²x. Rewriting it as a fraction over 1 = sin²x + cos²x first makes this easier.
  2. sin 2x = 2 sin x cos x ÷ (sin²x + cos²x).
  3. Dividing every term by cos²x turns sin x / cos x into tan x and cos²x / cos²x into 1.
  4. The result is sin 2x = 2 tan x / (1 + tan²x).

Use this form when a question gives you tan x.

Sin²x formula (sin square x)

Sin²x means (sin x)², the square of the sine. It is a different expression from sin 2x.

For example, sin²30° = (1/2)² = 1/4, but sin(2 × 30°) = √3/2.

There are two sin²x formulas:

\sin^2 x = 1 - \cos^2 x \qquad \sin^2 x = \frac{1 - \cos 2x}{2}

  • First form: subtract cos²x from both sides of sin²x + cos²x = 1.
  • Second form: rearrange cos 2x = 1 − 2sin²x to get 2sin²x = 1 − cos 2x, then divide by 2. This version is the one you need for integration.

Where the sin 2x formula is used

  • CBSE Class 11: trigonometric identities and simplifying expressions
  • A Level Maths: double angle identities, proofs and equations
  • IB Maths AA: identities, solving equations and calculus, covered in the IB section below
  • Calculus: the sin²x form makes integrals of sin²x possible

Solved examples on the sin 2x formula

Each example follows the same steps: write what's given, find any missing ratio, choose the formula, substitute, then simplify.

Example 1: cos A = 12/13 with A acute. What is sin 2A?

  • Given: cos A = 12/13
  • Missing ratio: 13² − 12² = 25, so the opposite side is 5 and sin A = 5/13 (positive, since A is acute)
  • Formula: sin 2A = 2 sin A cos A
  • Substitute: 2 × 5/13 × 12/13
  • Answer: sin 2A = 120/169

Example 2: sin A = 1/3 with A acute. What is sin 2A?

  • Given: sin A = 1/3
  • Missing ratio: cos²A = 1 − 1/9 = 8/9, so cos A = 2√2/3
  • Formula: sin 2A = 2 sin A cos A
  • Substitute: 2 × 1/3 × 2√2/3
  • Answer: sin 2A = 4√2/9

Example 3: Only tan A is known: tan A = 5/12. Find sin 2A

  • Given: tan A = 5/12
  • Formula: use the tan form, sin 2A = 2 tan A / (1 + tan²A)
  • Substitute: (10/12) ÷ (1 + 25/144) = (5/6) ÷ (169/144)
  • Simplify: 5/6 × 144/169 = 120/169
  • Answer: sin 2A = 120/169. It matches Example 1, because a 5–12–13 triangle sits behind both.

Example 4: sin A = 8/17 with A acute. Find both sin 2A and cos 2A

  • Given: sin A = 8/17, so cos A = 15/17 (an 8–15–17 triangle)
  • sin 2A: 2 × 8/17 × 15/17 = 240/289
  • cos 2A: cos²A − sin²A = 225/289 − 64/289 = 161/289

Example 5: Work out 2 sin 22.5° cos 22.5° with no calculator

  • Spot the pattern: this is 2 sin x cos x with x = 22.5°
  • Rewrite: 2 sin x cos x = sin 2x = sin 45°
  • Answer: √2/2

Example 6: Find the period of sin 2x

  • Rule: the period of sin bx is 2π / |b|
  • Substitute: b = 2, so the period = 2π / 2
  • Answer: π radians (180°)

For IB students: sin 2x in equations and calculus

This section is for IB Maths AA and A Level students. If you're studying CBSE, you can skip to the common mistakes.

Solving sin 2x = sin x for 0 ≤ x ≤ 2π

  • Rewrite: 2 sin x cos x − sin x = 0
  • Factorise: sin x (2 cos x − 1) = 0. Don't divide by sin x, or you lose solutions.
  • sin x = 0: x = 0, π, 2π
  • cos x = 1/2: x = π/3, 5π/3
  • Answer: x = 0, π/3, π, 5π/3, 2π

Solving sin 2x = 1/2 for 0 ≤ x ≤ 2π (the doubled range)

  • Doubled range: if 0 ≤ x ≤ 2π, then 0 ≤ 2x ≤ 4π
  • Solve for 2x: 2x = π/6, 5π/6, 13π/6, 17π/6
  • Halve: x = π/12, 5π/12, 13π/12, 17π/12
  • Students who forget to double the range find only two of the four answers.

Calculus with sin 2x and sin²x

  • d/dx (sin 2x) = 2 cos 2x, using the chain rule
  • ∫ sin 2x dx = −½ cos 2x + C
  • ∫ sin²x dx = ∫ (1 − cos 2x)/2 dx = x/2 − (sin 2x)/4 + C

See our Integration Formulas guide for more.

Common mistakes

  1. Writing sin 2x = 2 sin x. The angle doubles, not the value. The correct identity is 2 sin x cos x.
  2. Confusing sin 2x with sin²x. sin 2x is the sine of a doubled angle. sin²x is the sine squared.
  3. Using the wrong tan form. sin 2x = 2 tan x / (1 + tan²x). The minus version, 2 tan x / (1 − tan²x), is tan 2x.
  4. Choosing the wrong sign. When you find sin x or cos x from a square root, check the quadrant.
  5. Forgetting the doubled range when solving equations in 2x.
  6. Missing the chain rule. d/dx (sin 2x) = 2 cos 2x, not cos 2x.
  7. Forgetting + C when integrating.

Practice questions

  1. sin A = 7/25 and A is acute. Find sin 2A.
  2. cos A = 8/17 and A is acute. Find sin 2A.
  3. tan A = 1/2. Find sin 2A.
  4. Write sin²x using cos x only.
  5. Write sin²x using cos 2x.
  6. How long is one cycle of sin 2x, in radians?
  7. (IB) Integrate sin²x with respect to x.
  8. sin A = 20/29 and A is acute. Find sin 2A and cos 2A.

Answers

  1. cos A = 24/25, so sin 2A = 2 × 7/25 × 24/25 = 336/625
  2. sin A = 15/17, so sin 2A = 2 × 15/17 × 8/17 = 240/289
  3. 2(1/2) ÷ (1 + 1/4) = 4/5
  4. sin²x = 1 − cos²x
  5. sin²x = (1 − cos 2x)/2
  6. π
  7. x/2 − (sin 2x)/4 + C
  8. cos A = 21/29, so sin 2A = 840/841 and cos 2A = (441 − 400)/841 = 41/841

Frequently asked questions

What is the sin 2x formula?

sin 2x = 2 sin x cos x. In terms of tan, sin 2x = 2 tan x / (1 + tan²x).

Is sin 2x equal to 2 sin x?

No. Try x = 30°: sin 60° is about 0.866, while 2 sin 30° is exactly 1. The cosine factor in 2 sin x cos x is what makes the identity work.

What is sin 2A in terms of cos?

Replace sin A with ±√(1 − cos²A) in 2 sin A cos A to get sin 2A = ±2 cos A √(1 − cos²A). Pick the sign that matches the quadrant A lies in.

What is sin 2A in terms of sin?

Swap cos A for ±√(1 − sin²A), which gives sin 2A = ±2 sin A √(1 − sin²A). Again, the quadrant decides the sign.

What is sin 2x in terms of tan?

sin 2x = 2 tan x / (1 + tan²x). It's the quickest route whenever a question hands you tan x instead of sin x or cos x.

How do you prove the sin 2x formula?

Set both angles equal to x in the sine addition identity. The two products on the right are the same, so they combine into 2 sin x cos x.

What is the period of sin 2x?

One full cycle of sin 2x takes π radians (180°), half as long as sin x. In general, sin bx repeats every 2π/|b|.

What is the sin²x formula?

sin²x = 1 − cos²x, or sin²x = (1 − cos 2x)/2.

What is the difference between sin 2x and sin²x?

In sin 2x, the 2 doubles the angle before you take the sine. In sin²x, the 2 squares the answer after you take the sine. At 30° they give √3/2 and 1/4, which are very different.

What is the sin²A − sin²B formula?

sin²A − sin²B factorises neatly as sin(A + B) × sin(A − B). It is a useful shortcut in CBSE Class 11 identity proofs.

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