Double Angle Formula Calculator
Calculate sin(2x), cos(2x), and tan(2x) from an angle or from known trigonometric values, with formulas and step-by-step work.
Double angle identities for trigonometry
A double angle formula calculator evaluates trigonometric expressions where the angle is doubled. Enter an angle x, or enter known values such as sin(x), cos(x), or tan(x), and the calculator returns sin(2x), cos(2x), tan(2x), formulas, and decimal results.
The core formulas are sin(2x) = 2sin(x)cos(x), cos(2x) = cos^2(x) - sin^2(x), and tan(2x) = 2tan(x) / (1 - tan^2(x)). These identities are used in algebra, precalculus, calculus, physics, and graphing problems.
The double angle formula calculator solves trigonometric double angle equations for sine, cosine, and tangent. Enter an angle in degrees or radians to calculate the double angle result instantly.
Use angle mode for quick numeric answers, or value mode when a problem gives one or more trig values instead of the angle itself.
Double angle result
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sin(2x)
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cos(2x)
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tan(2x)
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Angle check
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Step-by-step work
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Note: Tangent is undefined when cos(2x) = 0 or when the denominator 1 - tan^2(x) equals zero. Rounding can make values near an undefined point look very large.
How to use the double angle formula calculator
- Choose an input method: Use angle mode if you know x, or value mode if the problem gives sin(x), cos(x), or tan(x).
- Enter the angle or trig values: In angle mode, enter x and choose degrees, radians, or pi radians. In pi radians mode, enter 1/12 for pi/12 or 3/4 for 3pi/4.
- Calculate: The calculator returns sin(2x), cos(2x), tan(2x), formulas, exact-value hints for common angles, and a short step-by-step explanation.
- Use the angle check: Review the doubled angle, normalized coterminal angle, quadrant, and reference angle before copying an answer.
- Check tangent warnings: If the tangent denominator is zero or nearly zero, tan(2x) may be undefined.
Double angle formulas
Double angle identities rewrite a trigonometric function of 2x in terms of functions of x. They come from the angle addition formulas by setting the two angles equal.
sin(2x) = 2sin(x)cos(x)
cos(2x) = cos^2(x) - sin^2(x) = 1 - 2sin^2(x) = 2cos^2(x) - 1
tan(2x) = 2tan(x) / (1 - tan^2(x))
The three cosine forms are equivalent because sin^2(x) + cos^2(x) = 1. Choose the form that matches the values given in the problem.
Formula reference: Wolfram MathWorld - Double-Angle Formulas.
Common double angle values
| x | 2x | sin(2x) | cos(2x) | tan(2x) |
|---|---|---|---|---|
| 15 degrees | 30 degrees | 1/2 | sqrt(3)/2 | sqrt(3)/3 |
| 30 degrees | 60 degrees | sqrt(3)/2 | 1/2 | sqrt(3) |
| 45 degrees | 90 degrees | 1 | 0 | Undefined |
| 60 degrees | 120 degrees | sqrt(3)/2 | -1/2 | -sqrt(3) |
Choosing the right identity
A double angle problem is usually easiest when you choose the formula that matches the information you already have. This prevents unnecessary square roots, sign mistakes, and quadrant confusion.
Given sin(x) and cos(x)
Use sin(2x) = 2sin(x)cos(x) and cos(2x) = cos^2(x) - sin^2(x).
Given only sin(x)
Use cos(2x) = 1 - 2sin^2(x). You may need quadrant information to determine cos(x) for sin(2x).
Given tan(x)
Use tan(2x) = 2tan(x) / (1 - tan^2(x)), and check whether the denominator is zero.
Exact value workflow for double angle problems
Many homework and test problems expect an exact answer instead of a decimal. Use the calculator to confirm the numeric value, then translate the result back into a familiar unit-circle value whenever the doubled angle is a special angle.
Step 1
Double the angle first
If x = 75 degrees, work with 2x = 150 degrees. The signs and exact values come from the doubled angle, not from the original angle alone.
Step 2
Use the reference angle
For 150 degrees, the reference angle is 30 degrees, so sin(150) = 1/2 and cos(150) = -sqrt(3)/2.
Step 3
Check the quadrant sign
A decimal like -0.8660 often points to -sqrt(3)/2. Use the quadrant of 2x to decide whether the exact value is positive or negative.
How double angle formulas help solve equations
Double angle formulas are often used to rewrite a trig equation so it contains one type of function. This makes factoring, substitution, and graph comparison much easier.
Equation has sin(x)
Use cos(2x) = 1 - 2sin^2(x) when you want everything in terms of sine.
Equation has cos(x)
Use cos(2x) = 2cos^2(x) - 1 when you want a cosine-only expression.
Equation has tan(x)
Use tan(2x) = 2tan(x)/(1 - tan^2(x)), then check for excluded values.
For example, cos(2x) = 1/2 can become 2cos^2(x) - 1 = 1/2, which is an algebraic equation in cos(x). After solving the algebra, always return to the original angle interval and check for all valid solutions.
Equation-solving reference: mathcentre - The double angle formulae.
Common mistakes and how to catch them
| Mistake | Why it happens | Quick check |
|---|---|---|
| Doubling the result instead of the angle | sin(2x) is not usually 2sin(x). The formula needs both sin(x) and cos(x). | For x = 30 degrees, sin(60) = 0.8660, not 1. |
| Using the wrong cosine form | All cosine forms are valid, but the best one depends on what values are known. | Given sin(x), use 1 - 2sin^2(x). Given cos(x), use 2cos^2(x) - 1. |
| Forgetting undefined tangent values | tan(2x) has a denominator, so some inputs create division by zero. | If tan(x) = 1 or -1, then tan(2x) is undefined. |
| Mixing degrees and radians | The same number means different angles depending on the unit mode. | 30 degrees is not 30 radians. Match the calculator unit to the problem statement. |
Trig formula review: Paul's Online Notes - Algebra/Trig Review: Trig Formulas.
Interesting fact
A full turn around a circle is 2pi radians, which is about 6.283185 radians, or 360 degrees. That means a doubled angle can wrap around the unit circle quickly: if x = 200 degrees, then 2x = 400 degrees, which is coterminal with 40 degrees. This is one reason double angle identities are so useful for simplifying periodic graphing and trigonometric equation problems. Source: Britannica - Radian.
Frequently Asked Questions
What is a double angle formula calculator in trigonometry?
A double angle formula calculator is a trigonometry tool that evaluates the sine, cosine, and tangent of a doubled angle. It uses a standard identity to transform an angle, theta expression, equation, or known trig value into a result for 2theta or 2x. This is useful when a triangle, unit-circle problem, or algebra step gives information about x but asks for a doubled-angle function.
Which double angle formula should I use for sine, cosine, or tangent?
Use sin(2x) = 2sin(x)cos(x) when the problem gives both sine and cosine values. Use cos(2x) = cos^2(x) - sin^2(x), 1 - 2sin^2(x), or 2cos^2(x) - 1 depending on which value is known. Use tan(2x) = 2tan(x) / (1 - tan^2(x)) when the expression or equation is already written with tangent.
When does the tangent double angle result have no solution?
The tangent result is undefined when cos(2x) equals zero, because tangent is sine divided by cosine. In the tangent double angle formula, this happens when 1 - tan^2(x) = 0, so tan(x) = 1 or tan(x) = -1. The calculator will report an undefined value instead of forcing a decimal solution where the function does not exist.
Can I enter an angle in degree or radian measure?
Yes. In angle mode, choose degree or radian input to match your worksheet, calculator setting, or textbook notation. The formulas do not change with the unit; only the numeric interpretation of the angle changes. For example, 30 degrees and 30 radians are very different inputs, so matching the unit is essential for a correct result.
Why does the cosine double angle identity have multiple forms?
The cosine double angle identity has multiple forms because the Pythagorean identity lets you replace sin^2(x) with 1 - cos^2(x), or replace cos^2(x) with 1 - sin^2(x). Each formula gives the same cosine value, but one form may make the algebra cleaner than the others. That flexibility is why double angle formulas often appear in simplifying an expression, solving a trigonometric equation, or preparing a function for graphing.
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Disclaimer: This double angle formula calculator is for general educational and informational use only. It provides trigonometry calculations based on user-entered values and rounded numerical output. It may not match every classroom convention, textbook notation, exact-value requirement, proof method, or assessment format. Always verify the required form of your answer with your teacher, textbook, course instructions, or official solution key.
Last updated: May 21, 2026