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Double Angle Formulas: Sin, Cos, Tan Identities

TL;DR

The double angle formulas are sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ − sin²θ (also 1 − 2 sin²θ and 2 cos²θ − 1), and tan 2θ = 2 tan θ / (1 − tan²θ). All three are derived from the sum formulas by setting α = β = θ. The cosine has three equivalent forms because the Pythagorean identity converts between them.

The Double Angle Formulas

The double angle formulas are a set of three trigonometric identities that express sin 2θ, cos 2θ, and tan 2θ in terms of sin θ, cos θ, and tan θ. The cosine formula has three equivalent forms. Each identity also has a t-form expressed purely in terms of tan θ.

Double Angle Formulas at a Glance

Variable Key

A common notation trap: sin²θ means (sin θ)², not sin(θ²). The two are completely different quantities.

The Three Double Angle Formulas

Double Angle Formula for Sine — sin 2θ

sin 2θ = 2 sin θ cos θ

The sine of a double angle equals twice the product of sin θ and cos θ. The t-form is:

sin 2θ = 2 tan θ / (1 + tan²θ)

The t-form is useful when only tan θ is known.

Double Angle Formulas for Cosine — cos 2θ (Three Forms)

The cosine of a double angle has three standard forms:

cos 2θ = cos²θ − sin²θ
cos 2θ = 1 − 2 sin²θ
cos 2θ = 2 cos²θ − 1

The second form follows from the first by replacing cos²θ with 1 − sin²θ. The third form follows from the first by replacing sin²θ with 1 − cos²θ. The t-form is:

cos 2θ = (1 − tan²θ) / (1 + tan²θ)

Double Angle Formula for Tangent — tan 2θ

tan 2θ = 2 tan θ / (1 − tan²θ)

This formula is undefined when tan²θ = 1, that is, when θ = 45° + 90°·n for any integer n. At those angles, 2θ is an odd multiple of 90°, where tan is undefined.

Derivation of Double Angle Formulas

The double angle formulas are special cases of the sum formulas, obtained by substituting α = β = θ.

Deriving sin 2θ from the Sine Sum Formula

Start with the sine sum formula:

sin(α + β) = sin α cos β + cos α sin β

Set α = β = θ:

sin(θ + θ) = sin θ cos θ + cos θ sin θ
sin 2θ = 2 sin θ cos θ

Deriving cos 2θ and Its Three Forms

Start with the cosine sum formula:

cos(α + β) = cos α cos β − sin α sin β

Set α = β = θ:

cos 2θ = cos²θ − sin²θ

Apply the Pythagorean identity sin²θ + cos²θ = 1 in two ways. Replacing cos²θ with 1 − sin²θ gives:

cos 2θ = (1 − sin²θ) − sin²θ = 1 − 2 sin²θ

Replacing sin²θ with 1 − cos²θ gives:

cos 2θ = cos²θ − (1 − cos²θ) = 2 cos²θ − 1

Deriving tan 2θ from the Tangent Sum Formula

Start with the tangent sum formula:

tan(α + β) = (tan α + tan β) / (1 − tan α · tan β)

Set α = β = θ:

tan 2θ = (tan θ + tan θ) / (1 − tan θ · tan θ) = 2 tan θ / (1 − tan²θ)

The t-Formulas: sin 2θ, cos 2θ, tan 2θ in Terms of tan θ

When only tan θ is known, all three double angle identities can be written using t = tan θ:

These forms are derived using the identity sec²θ = 1 + tan²θ. For example, starting from sin 2θ = 2 sin θ cos θ and dividing both numerator and denominator by cos²θ gives 2 tan θ / sec²θ, which simplifies to 2t / (1 + t²).

When to Use Double Angle Formulas

Double angle formulas appear in several standard situations:

• Simplifying expressions where 2θ appears alongside θ
• Solving trigonometric equations that mix single and double angles, such as cos 2x = sin x
• Verifying trigonometric identities
• Deriving reduction formulas, half-angle formulas, and triple-angle formulas
• Computing exact values (for example, sin 60° from sin 30° and cos 30°)
• Integrating expressions involving sin²x or cos²x via power reduction

Common Mistakes with Double Angle Formulas

A short list of recurring errors:

• Confusing sin²θ with sin 2θ. The first means (sin θ)². The second is the sine of twice the angle. Numerically very different.
• Ignoring the quadrant when finding the missing function. If only cos θ is given, sin θ has two possible signs; the quadrant determines which.

Table of Derived Values Using Double Angle Formulas

The double angle formulas convert known trigonometric values at 30°, 45°, and 60° into values at 60°, 90°, and 120°:

Hyperbolic Double-Angle Formulas (Brief Note)

The hyperbolic counterparts mirror the trigonometric forms with one sign difference in the tanh denominator:

sinh 2x = 2 sinh x cosh x
cosh 2x = cosh²x + sinh²x = 1 + 2 sinh²x = 2 cosh²x − 1
tanh 2x = 2 tanh x / (1 + tanh²x)

Frequently Asked Questions

What are the double angle formulas? The double angle formulas are sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ − sin²θ (with two equivalent alternate forms), and tan 2θ = 2 tan θ / (1 − tan²θ). They express trigonometric functions of 2θ in terms of functions of θ.

How do you derive the double angle formulas? Start with the sum formulas - sin(α + β), cos(α + β), tan(α + β) - and substitute α = β = θ. This collapses the sum into a function of 2θ.

Why does cos 2θ have three forms? The base form is cos 2θ = cos²θ − sin²θ. Using sin²θ + cos²θ = 1, you can replace either cos²θ or sin²θ with the other to get 1 − 2 sin²θ or 2 cos²θ − 1. All three are mathematically identical; the choice depends on which trigonometric value is known or which form simplifies the next step.

What is the difference between double angle formulas and half-angle formulas? Double angle formulas express functions of 2θ in terms of functions of θ. Half-angle formulas do the reverse - they express functions of θ/2 in terms of functions of θ.