Double Angle Calculator - Sine, Cosine, and Tangent of 2θ

A double angle calculator is a trigonometry tool that takes a single angle θ and returns the trig values at the doubled angle 2θ in one step. Give it any input angle in degrees or radians and it produces sin(2θ), cos(2θ), and tan(2θ) together with the doubled angle itself in both units. Students and engineers reach for it whenever a problem gives a known angle and asks for a derived one, whether the source is a half-angle substitution or a triangle whose sides lead to a doubled reference angle.

• • Solving double-angle identity problems: Check homework that asks you to evaluate sin(2θ), cos(2θ), or tan(2θ) for a specific θ without rebuilding the formula on paper.
• • Verifying hand derivations: Compare a manual application of sin(2θ) = 2 sin(θ) cos(θ) or cos(2θ) = cos²(θ) − sin²(θ) with a numeric result in seconds.
• • Working with half-angle substitutions: Compute 2θ from a known θ, then evaluate the trig values at the new angle as part of a power-reduction workflow.

Pick the unit that matches the rest of your work and the calculator handles every conversion internally. It then reports 2θ in both degrees and radians so you do not have to re-convert before plugging the answer into the next step of the problem.

When you need the inverse direction of cosine, where you already know cos(θ) and want the angle, the Arccos Calculator handles principal-angle recovery without leaving the math-conversion cluster.

The calculator reads the original angle and the unit selector, converts the angle to a single radian representation, and then evaluates the three double-angle identities at 2θ. Every output is computed from the doubled angle, and the identity check is the right-hand side of the sine identity, which should match sin(2θ) to floating-point precision.

• θ (theta): The original angle you enter. The unit selector tells the calculator whether it is in degrees or radians.
• 2θ: The doubled angle, returned in degrees and radians. Every trig output below is evaluated at this angle.
• sin(2θ), cos(2θ), tan(2θ): The three trig values at 2θ, computed directly from the doubled angle and cross-checked with the right-hand sides of the identities.

The identity check on the results panel is a numeric verification of sin(2θ) = 2 sin(θ) cos(θ). When the two numbers match to four decimal places, the rest of the outputs are trustworthy for the same input.

According to Wolfram MathWorld: Double-Angle Formulas, the three double-angle identities sin(2θ) = 2 sin(θ) cos(θ), cos(2θ) = cos²(θ) − sin²(θ), and tan(2θ) = 2 tan(θ) / (1 − tan²(θ)) follow from the angle addition identities with α = β = θ

The right-hand side 2 sin(θ) cos(θ) is just a restatement of the sine addition identity, and the Arcsin Calculator handles the inverse direction for sine when you only have a sine value in hand.

Four concepts make the rest of the calculator easy to read. They are the same building blocks the identities are derived from.

These four ideas explain why the same 2θ value can produce very different output rows for sine, cosine, and tangent, and why the calculator surfaces an explicit 'undefined' for the tangent case instead of a misleading huge number.

The tangent side of the double-angle identity is the one that goes undefined most often, so the Arctan Calculator is a useful companion when you need to recover a principal angle from a tangent value.

Working with the calculator takes a few seconds. Pick the unit, type the angle, and read the doubled angle plus all three trig values plus the identity check from the results panel.

1. 1 Pick the unit: Choose whether the angle you are about to type is in degrees or radians. The calculator does the conversion internally.
2. 2 Enter the original angle θ: Type any numeric angle, including negative values and angles outside 0 to 360. The default is 30, the cleanest reference case.
3. 3 Read the doubled angle: The first two results show 2θ in degrees and radians, so you can drop the doubled angle straight into the rest of the problem.
4. 4 Read sin(2θ), cos(2θ), tan(2θ): The three trig values at the doubled angle appear in the same order as the identities, with tan(2θ) shown as 'undefined' whenever cos(2θ) is too close to zero.
5. 5 Verify with the identity check: Compare the right-hand side 2 sin(θ) cos(θ) with sin(2θ). The two numbers should match to four decimal places for any well-behaved input.

Suppose a half-angle problem gives θ = 22.5° and asks for sin(2θ), cos(2θ), and tan(2θ). Enter 22.5 with degrees selected, read 2θ = 45° (π/4 rad) at the top, then read sin(2θ) ≈ 0.7071, cos(2θ) ≈ 0.7071, and tan(2θ) = 1. The identity check at the bottom shows 2 sin(22.5°) cos(22.5°) ≈ 0.7071, which matches sin(2θ).

If you need to flip between degrees and radians outside the calculator, the Angle Converter switches the unit without changing the underlying value, which is handy when the same problem mixes both.

A single screen that does the unit conversion, applies the three identities, and runs a numeric check saves time on homework, teaching prep, and code reviews.

• • All three identities in one view: sin(2θ), cos(2θ), and tan(2θ) appear together, so a problem that asks for all three is one input away from a full answer.
• • Both angle units at once: The doubled angle shows in degrees and radians, which removes the most common cause of off-by-factor errors when moving between trig problems and formula sheets.
• • Built-in identity check: The right-hand side 2 sin(θ) cos(θ) sits next to sin(2θ) so the identity is visible, not just assumed.
• • Undefined tangent is handled cleanly: The calculator reports 'undefined' for tan(2θ) when cos(2θ) is essentially zero, instead of showing a misleading overflow number.
• • Negative and out-of-range angles work: Inputs outside 0 to 360 and negative angles evaluate correctly because the calculator uses the same Math.sin, Math.cos, and Math.tan functions that the identities are written for.

The biggest practical win is keeping the original angle, the doubled angle, and the trig values on the same screen, which makes the relationships stay visible at every step.

The double-angle formulas sit on top of basic sine and cosine ratios, so the Right Triangle Calculator is a useful companion for triangle problems where the same ratios already drive the side and angle work.

Most of the time the calculator returns the expected numbers, but a handful of factors control what the tool can and cannot do. Knowing them up front prevents the most common mistakes, especially around the unit selector and the tangent branch.

• • The tool returns the principal real values of sin(2θ), cos(2θ), and tan(2θ). It does not compute complex-valued trig outputs for inputs that fall outside the standard range, because that is rarely what classroom or applied problems need.
• • Floating-point arithmetic means the identity check matches sin(2θ) only to roughly 15 significant digits. Treat the identity check as a sanity check, not an exact equality test, especially for large angles.

If a problem demands an exact symbolic answer, work the identities by hand and use the calculator to confirm the numerical value. If it demands a numerical answer, the calculator is enough on its own.

According to Wikipedia: List of trigonometric identities, the double-angle formulas are special cases of the angle addition identities with both summands equal to the same angle

Central-angle problems also rely on doubled angles like 2θ, so the Central Angle Calculator keeps arc length, chord length, and sector area in the same workflow as the trig values the double-angle identities produce.