What is the double angle formula for \tan 2\theta?

The double angle formula for tangent is \tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta}, which you can derive by dividing the sine double angle formula by the cosine double angle formula. It is useful when you know \tan \theta and need \tan 2\theta.

Why does the cosine double angle formula have multiple forms like 1 - 2\sin^2 \theta and 2\cos^2 \theta - 1?

These are all equivalent forms of the same formula because of the Pythagorean Identity \sin^2 \theta + \cos^2 \theta = 1. You can substitute \cos^2 \theta = 1 - \sin^2 \theta (or vice versa) into \cos 2\theta = \cos^2 \theta - \sin^2 \theta to derive the other forms. Use whichever form matches what you already know.

Do double angle formulas work the same way with degrees as with radians?

Yes, the double angle formulas work identically whether \theta is in degrees or radians. The formulas \sin 2\theta = 2\sin \theta \cos \theta and \cos 2\theta = \cos^2 \theta - \sin^2 \theta are pure algebraic relationships that do not depend on the unit of angle measurement.

How do I know which form of the cosine double angle formula to use?

It depends on what you already know. If you know both \sin \theta and \cos \theta, use \cos^2 \theta - \sin^2 \theta. If you only know \sin \theta, use 1 - 2\sin^2 \theta. If you only know \cos \theta, use 2\cos^2 \theta - 1. Picking the form that matches your given information makes the arithmetic much faster.

Double Angle Formulas

Double angle formulas express trig functions of 2θ in terms of single angles, letting you simplify complex expressions and solve trigonometric equations.

\sin 2\theta = 2\sin\theta\cos\theta, \quad \cos 2\theta = \cos^2\theta - \sin^2\theta

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What each symbol means

What Double Angle Formulas takes

Symbol	Meaning
$\theta$	The original angle, in radians or degrees; it is the base angle whose double, $2\theta$ , appears in the formulas—confusing which is which will reverse your calculation.

When to use it

Use these formulas whenever you encounter an expression or equation with $\sin 2\theta$ or $\cos 2\theta$ and you need to work with $\sin \theta$ and $\cos \theta$ instead.

Level

Usually taught in: Pre-Calculus

Worked examples

1. Simplify sin(2θ) with direct values

Problem

Find

\sin 2\theta

\sin \theta = \frac{1}{2}

and

\cos \theta = \frac{\sqrt{3}}{2}

$\sin 2\theta = 2\sin \theta \cos \theta$
Start with the double angle formula for sine.
$\sin 2\theta = 2 \cdot \frac{1}{2} \cdot \frac{\sqrt{3}}{2}$
Substitute the given values of $\sin \theta$ and $\cos \theta$ into the formula.
$\sin 2\theta = \frac{2\sqrt{3}}{4} = \frac{\sqrt{3}}{2}$
Multiply the fractions: $2 \cdot \frac{1}{2} \cdot \frac{\sqrt{3}}{2} = \frac{2\sqrt{3}}{4}$ , then simplify by canceling the 2.

Answer: $\sin 2\theta = \frac{\sqrt{3}}{2}$

The double angle formula for sine is direct: when you know both $\sin \theta$ and $\cos \theta$ , you multiply them together (with the factor of 2) to get $\sin 2\theta$ immediately. This is faster than other methods.

2. Find cos(2θ) with negative values and fractions

Problem

Find

\cos 2\theta

\sin \theta = -\frac{3}{5}

and

\cos \theta = -\frac{4}{5}

$\cos 2\theta = \cos^2 \theta - \sin^2 \theta$
Apply the double angle formula for cosine.
$\cos^2 \theta = \left( -\frac{4}{5} \right)^2 = \frac{16}{25}$
Square $\cos \theta$ ; remember that $(-4)^2 = 16$ even though $\cos \theta$ is negative.
$\sin^2 \theta = \left( -\frac{3}{5} \right)^2 = \frac{9}{25}$
Square $\sin \theta$ the same way; the negative sign becomes positive when you square.
$\cos 2\theta = \frac{16}{25} - \frac{9}{25} = \frac{7}{25}$
Subtract to get the final answer.

Answer: $\cos 2\theta = \frac{7}{25}$

This example shows why squaring is crucial: both $\sin \theta$ and $\cos \theta$ are negative, but their squares are both positive, so the subtraction works as expected. The formula still produces the correct answer regardless of the signs of the original values.

3. Cyclist's elevation gain on a sloped path

Problem

A cyclist rides along a hill whose elevation relative to the starting point is

h(\theta) = 30(1 - \cos 2\theta)

meters, where

\theta

is the angle along the slope. At one point,

\sin \theta = \frac{1}{2}

and

\cos \theta = \frac{\sqrt{3}}{2}

. Calculate the elevation

h

at that location.

$\cos^2 \theta = \left( \frac{\sqrt{3}}{2} \right)^2 = \frac{3}{4}$
Square the cosine value to prepare for the double angle formula; $(\sqrt{3})^2 = 3$ .
$\sin^2 \theta = \left( \frac{1}{2} \right)^2 = \frac{1}{4}$
Square the sine value.
$\cos 2\theta = \frac{3}{4} - \frac{1}{4} = \frac{2}{4} = \frac{1}{2}$
Apply the double angle formula for cosine by subtracting the squared values and simplify.
$h(\theta) = 30 \left( 1 - \frac{1}{2} \right) = 30 \cdot \frac{1}{2} = 15$
Substitute $\cos 2\theta = \frac{1}{2}$ into the height model to find the elevation at this angle.

Answer: $h(\theta) = 15 \text{ meters}$

Real-world elevation and wave models often depend on double angle expressions. The double angle formula converts simple angle measurements ( $\sin \theta$ , $\cos \theta$ ) into the doubled-angle behavior ( $\cos 2\theta$ ) that describes the actual height, making these formulas indispensable for applied problems.

Common mistakes

Where Double Angle Formulas usually goes wrong

The mistake	Why it is wrong	The fix
Writing $\sin 2\theta = 2\sin \theta + 2\cos \theta$ instead of a product.	Students sometimes misread the formula and treat the 2 as something to distribute to a sum, rather than as a coefficient on the product $\sin \theta \cos \theta$ .	The double angle formula for sine is always $\sin 2\theta = 2 \sin \theta \cos \theta$ —it is a product of sine and cosine, not a sum.
Confusing the formula and writing $\cos 2\theta = \cos^2 \theta + \sin^2 \theta$ , which always equals 1.	This looks similar to the Pythagorean identity $\cos^2 \theta + \sin^2 \theta = 1$ , but the double angle formula uses subtraction, not addition.	Remember that $\cos 2\theta = \cos^2 \theta - \sin^2 \theta$ , with a minus sign, not a plus.
Forgetting to square the sine and cosine before substituting them into $\cos 2\theta = \cos^2 \theta - \sin^2 \theta$ .	Students may substitute the raw values and write $\cos 2\theta = \frac{\sqrt{3}}{2} - \frac{1}{2}$ instead of computing $\cos^2 \theta$ and $\sin^2 \theta$ first.	Always calculate $\cos^2 \theta = (\cos \theta)^2$ and $\sin^2 \theta = (\sin \theta)^2$ before you substitute them into the formula.

Tips and when to use something else

The cosine double angle formula has two other equivalent forms: $\cos 2\theta = 1 - 2\sin^2 \theta$ and $\cos 2\theta = 2\cos^2 \theta - 1$ , which come from the Pythagorean Identity; choose whichever form matches the information you already have.
If you only know one of $\sin \theta$ or $\cos \theta$ , use the Pythagorean Identity to find the other before applying the double angle formula.
For double angles of tangent, you can derive $\tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta}$ by dividing the sine formula by the cosine formula; use this when working with tangent.
If a problem feels complicated, check whether the Half Angle Formulas or the Addition Formulas might be simpler or more direct for your particular equation.

Related formulas

Sine Addition FormulaThe Sine Addition Formula expands sine of a sum or difference into a product of sines and cosines, making it useful for exact angle calculations.Cosine Addition FormulaFind the cosine of an angle sum or difference by using the cosine and sine values of the individual angles—essential for solving trigonometric equations.Pythagorean IdentityThe Pythagorean Identity states that sine squared plus cosine squared always equals one, helping you simplify trigonometric expressions and solve equations.Half Angle FormulasHalf angle formulas express trigonometric functions of half an angle in terms of the full angle, helping solve equations and simplify expressions.Unit CircleThe unit circle is a fundamental tool that shows how angles map to sine and cosine coordinates: the point (cos θ, sin θ) always lies on x² + y² = 1.

Frequently asked questions

What is the double angle formula for $\tan 2\theta$ ?: The double angle formula for tangent is $\tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta}$ , which you can derive by dividing the sine double angle formula by the cosine double angle formula. It is useful when you know $\tan \theta$ and need $\tan 2\theta$ .
Why does the cosine double angle formula have multiple forms like $1 - 2\sin^2 \theta$ and $2\cos^2 \theta - 1$ ?: These are all equivalent forms of the same formula because of the Pythagorean Identity $\sin^2 \theta + \cos^2 \theta = 1$ . You can substitute $\cos^2 \theta = 1 - \sin^2 \theta$ (or vice versa) into $\cos 2\theta = \cos^2 \theta - \sin^2 \theta$ to derive the other forms. Use whichever form matches what you already know.
Do double angle formulas work the same way with degrees as with radians?: Yes, the double angle formulas work identically whether $\theta$ is in degrees or radians. The formulas $\sin 2\theta = 2\sin \theta \cos \theta$ and $\cos 2\theta = \cos^2 \theta - \sin^2 \theta$ are pure algebraic relationships that do not depend on the unit of angle measurement.
How do I know which form of the cosine double angle formula to use?: It depends on what you already know. If you know both $\sin \theta$ and $\cos \theta$ , use $\cos^2 \theta - \sin^2 \theta$ . If you only know $\sin \theta$ , use $1 - 2\sin^2 \theta$ . If you only know $\cos \theta$ , use $2\cos^2 \theta - 1$ . Picking the form that matches your given information makes the arithmetic much faster.

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Reviewed 2026-09-18