Double Angle Formula Calculator
Evaluate sin(2θ), cos(2θ), and tan(2θ) from one angle, with degree, radian, and π-radian input.
Angle input
Use a decimal or a simple fraction such as 1/12. Commas and scientific notation are rejected.
In π-radian mode, 1 means π rad = 180°, so 1/12 means π/12.
This changes only the explanatory steps; it does not change the calculation.
Live results
Identity values
| Function | Double-angle identity | Current value |
|---|---|---|
| Double sine | sin(2θ) = 2 sin(θ) cos(θ) | 0.500000 |
| Double cosine | cos(2θ) = cos²(θ) – sin²(θ) | 0.866025 |
| Double tangent | tan(2θ) = 2 tan(θ) / (1 – tan²(θ)) | 0.577350 |
How to use this double angle formula calculator
What this calculator does
This calculator takes one angle θ and evaluates the three standard double-angle identities: sin(2θ), cos(2θ), and tan(2θ). It also converts the entered angle to degrees and radians so you can verify that the unit interpretation is correct. The results are mathematical identities, not numerical approximations to a physical measurement. The only approximation is the decimal display, which is rounded to six decimal places while the internal model and Excel export retain the underlying numeric value.
When to use it
Use it when you are simplifying a trigonometric expression, checking a homework derivation, converting an angle before applying an identity, or verifying the numerical value of a doubled angle. It is especially useful when an angle is supplied as a multiple of π, because you can enter a value such as 1/12 instead of typing a decimal approximation of π/12. For a formal review of these identities and their derivation from angle-sum formulas, see the OpenStax section on double-angle formulas.
How to calculate
- The calculator opens with a complete demonstration: Angle θ is 1/12, Angle unit is π radians, and Derivation is enabled. The example workbook is immediately available through Download Excel.
- Replace Angle θ with your value. A decimal such as 15, 0.25, or -2.5 is accepted, as is a simple fraction such as 1/12. Then choose Degrees (°), Radians (rad), or π radians.
- Read Double angle 2θ, Double sine, Double cosine, and Double tangent. The identity table repeats the same model values beside the formulas. Turn Derivation off if you only need the results.
- Choose Download Excel to export the current validated inputs and results as a real XLSX workbook. Reset clears the demonstration and all calculated content; the export button then stays disabled until a complete valid angle is entered again.
Input guide
Angle θ is required. It accepts a finite decimal or one simple fraction with a nonzero denominator. In Degrees (°), a value of 15 means 15°. In Radians (rad), 0.261799 is about 15°. In π radians, 1/12 means π/12. Values with commas, scientific notation, multiple slashes, or a zero denominator are rejected rather than silently reinterpreted. The calculator applies a generous finite magnitude limit for numerical reliability. Angle unit is required. When you change the unit, the calculator converts the current valid angle so the represented angle stays the same; for example, 15 degrees becomes about 0.261799 radians or 0.083333333333 π radians. For background on degree/radian conversion, consult OpenStax on angle measure and radian-degree conversion. Derivation is an optional display control: it shows or hides substitution lines and does not affect the model or workbook values.
Output guide
Double angle 2θ is exactly twice the supplied angle and is displayed in the selected unit. The summary pills also show θ in degrees, 2θ in degrees, and θ in radians. Double sine is sin(2θ) and always lies from – 1 to 1. Double cosine is cos(2θ) and also lies from – 1 to 1. Double tangent is tan(2θ); it can be positive or negative and becomes undefined when cos(2θ) is zero. Near those singular angles, tangent can change magnitude very rapidly, so the calculator reports an exact “undefined” state at the numerical singularity instead of displaying Infinity. These are exact trigonometric relationships evaluated numerically, and the unit-circle interpretation of sine and cosine is reviewed in OpenStax's unit-circle treatment of sine and cosine.
Worked example
The startup example uses θ = π/12, which is 15°. Doubling gives 2θ = π/6 = 30°. Then sin(2θ) = sin(30°) = 0.500000; cos(2θ) = cos(30°) = 0.866025; and tan(2θ) = tan(30°) = 0.577350. Using the identity form gives the same result: sin(2θ) = 2 sin(15°) cos(15°) = 0.5. The first-open controls, result cards, identity table, and exported workbook all use this same demonstration state.
How the double-angle identities work
The sine identity comes directly from the sine addition rule after setting both angles equal to θ: sin(θ + θ) = sin θ cos θ + cos θ sin θ, which simplifies to 2 sin θ cos θ. The cosine addition identity similarly yields cos(2θ) = cos² θ – sin² θ. Combining that result with sin² θ + cos² θ = 1 produces the equivalent forms 2 cos² θ – 1 and 1 – 2 sin² θ. The tangent identity follows either from the tangent addition rule or by dividing the double-angle sine by the double-angle cosine when the denominator is nonzero.
Common interpretation mistakes
A double angle means evaluating the trigonometric function at twice the angle; it does not mean doubling the function value. In general, sin(2θ) is not 2 sin θ, and cos(2θ) is not 2 cos θ. Another frequent mistake is mixing degrees and radians: the numeric text “1” means 1°, 1 radian, or π radians depending on the selected unit, and those are very different angles. Finally, tangent deserves special care because it is undefined at odd multiples of 90° for the doubled angle. When a problem lands on such a point, “undefined” is the correct mathematical outcome.