Sin2x Formula

Sin2x Formula (sin 2x = 2 sin x cos x)

The sin2x formula is one of the most important double angle identities in trigonometry. It expresses the sine of a doubled angle in terms of the sine and cosine of the original angle, and shows up constantly in CBSE board exams, JEE and NEET.

Class: 10 & 11
Topic: Trigonometric Identities
Exams: CBSE · JEE · NEET

What is the Sin2x Formula?

The sin2x formula gives the value of sin 2x — the sine of double the angle x — without you having to compute the doubled angle first. It belongs to the family of double angle identities in trigonometry.

The standard form of the sin 2x formula is:

Sin2x Formula
sin 2x = 2 sin x cos x

Here x can be any angle, measured in degrees or radians. Because it links a single trig ratio (sin 2x) to a product of two (sin x and cos x), it is one of the most frequently used identities for simplifying expressions and proving other results.

✓ Quick read
To find sin 2x, just multiply sin x, cos x and 2 together. You never need the actual value of the angle 2x.

Sin2x in Terms of Tan

When a problem only gives you the value of tan x, it is far quicker to use the sin2x in terms of tan form instead of finding sine and cosine separately:

Sin2x in terms of tan
sin 2x = 2 tan x1 + tan2x

This is algebraically identical to 2 sin x cos x — it is simply rewritten using only the tangent ratio. It is especially handy in integration, in the half-angle substitution, and in JEE problems where tan x is the given quantity.

Derivation of the Sin2x Formula

The sin 2x formula is derived directly from the angle sum identity for sine. Recall that:

sin(A + B) = sin A cos B + cos A sin B

Since 2x = x + x, we substitute A = B = x:

sin 2x = sin(x + x)Write the doubled angle as a sum.
= sin x cos x + cos x sin xApply the angle sum identity with A = B = x.
= 2 sin x cos xCombine the two identical terms.
Result
∴ sin 2x = 2 sin x cos x

Sin2x in Terms of Tan — Proof

Starting from sin 2x = 2 sin x cos x, we use the Pythagorean identity sin2x + cos2x = 1 and divide through by cos2x:

sin 2x = 2 sin x cos xsin2x + cos2xReplace 1 in the denominator with sin²x + cos²x.
= divide numerator and denominator by cos2xEvery term is scaled by 1 / cos²x.
= 2 tan xtan2x + 1Since sin x / cos x = tan x.
Result
∴ sin 2x = 2 tan x1 + tan2x

The sin2x formula is usually learnt alongside the other double angle identities. Here is the full set you should memorise for exams:

Ratio Standard form In terms of tan x
sin 2x 2 sin x cos x 2 tan x1 + tan2x
cos 2x cos2x − sin2x = 1 − 2sin2x = 2cos2x − 1 1 − tan2x1 + tan2x
tan 2x — 2 tan x1 − tan2x

A closely related rearrangement of the sin2x formula gives the power-reduction result, often used in integration:

sin2x = 1 − cos 2x2

Sin2x Value Table

Substituting standard angles into sin 2x = 2 sin x cos x gives the following values. Notice that the answer always equals sine of the doubled angle.

x 2x sin 2x Decimal
0° 0° 0 0
15° 30° 1/2 0.5
30° 60° √3 / 2 0.866
45° 90° 1 1
60° 120° √3 / 2 0.866
90° 180° 0 0

Common Mistakes to Avoid

× sin 2x ≠ 2 sin x
The most common error is dropping the cosine. sin 2x is not double sin x. The correct identity is always sin 2x = 2 sin x cos x.
× Watch the quadrant
When you find cos x from sin x using cos x = ±√(1 − sin2x), choose the correct sign based on the quadrant of x before plugging into the formula.

Solved Examples on Sin2x Formula

Example 1

If sin x = 3/5 and x is acute, find sin 2x.

Since x is acute, cos x = √(1 − 9/25) = 4/5.
sin 2x = 2 sin x cos x = 2 × (3/5) × (4/5)
sin 2x = 24/25

Example 2

If tan x = 5/12, find sin 2x using the tan form.

sin 2x = (2 tan x) / (1 + tan2x) = (2 × 5/12) / (1 + 25/144) = (10/12) / (169/144)
sin 2x = 120/169

Example 3

Find the value of sin 2x when x = 30°.

sin 2x = 2 sin 30° cos 30° = 2 × (1/2) × (√3/2)
sin 60° = √3/2 ≈ 0.866

Example 4

Prove that (1 − cos 2x) / sin 2x = tan x.

LHS = (1 − cos 2x) / sin 2x = (2 sin2x) / (2 sin x cos x)
using cos 2x = 1 − 2sin2x and sin 2x = 2 sin x cos x
= sin x / cos x = tan x = RHS
Hence proved

Practice Questions

Try these on your own to lock in the sin2x formula:

  1. If sin x = 8/17 and x is acute, find sin 2x. (Ans: 240/289)
  2. If tan x = 3/4, find sin 2x. (Ans: 24/25)
  3. Evaluate sin 2x at x = 45°. (Ans: 1)
  4. Express sin 4x using the sin2x formula. (Hint: treat 4x as 2·(2x))

Frequently Asked Questions

What is the sin2x formula?
The sin2x formula is sin 2x = 2 sin x cos x. It is a double angle identity that expresses the sine of twice an angle in terms of the sine and cosine of the original angle.
What is sin2x in terms of tan x?
In terms of tangent, sin 2x = 2 tan x / (1 + tan²x). This form is convenient when only the value of tan x is known.
Is sin2x equal to 2 sin x?
No. Writing sin 2x = 2 sin x is a common mistake. The cosine factor must be included: sin 2x = 2 sin x cos x.
How do you derive the sin2x formula?
Use the angle sum identity sin(A + B) = sin A cos B + cos A sin B and put A = B = x. This gives sin(x + x) = sin x cos x + cos x sin x = 2 sin x cos x.
What is the value of sin2x when x = 30°?
When x = 30°, sin 2x = sin 60° = √3/2 ≈ 0.866. Using the formula: 2 sin 30° cos 30° = 2 × (1/2) × (√3/2) = √3/2.


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