Double Angle Formula Calculator - Sine, Cosine, and Tangent 2x Identities

A double angle formula is a set of three trigonometric identities that express the trig values of a doubled angle 2θ in terms of the trig values of the original angle θ. The full set is sin(2θ) = 2 sin(θ) cos(θ), cos(2θ) = cos²(θ) − sin²(θ), and tan(2θ) = 2 tan(θ) / (1 − tan²(θ)). This calculator takes any θ in degrees or radians and returns 2θ plus sin(2θ), cos(2θ), and tan(2θ) in a single step, with a numeric identity check that confirms 2 sin(θ) cos(θ) agrees with sin(2θ).

• • Homework that asks you to apply the formulas: Plug in θ, read sin(2θ), cos(2θ), and tan(2θ) in one step and compare with the textbook answer.
• • Verifying a symbolic derivation: Cross-check a manual application of sin(2θ) = 2 sin(θ) cos(θ) or cos(2θ) = cos²(θ) − sin²(θ) against the numeric output.
• • Building up to power-reduction or half-angle work: Recover the formula values for 2θ so you can invert them into half-angle expressions and power-reduction substitutions.
• • Tracing why tan(2θ) goes undefined at 2θ = 90°: See the tangent branch break in real time at θ = 45°, the cleanest classroom example of a tangent asymptote from the formula.

The page-level formula box and the results panel work together: the formula box is the source of truth, and the results panel tells you whether the identity still holds for the angle you chose.

If you only need the numeric output for a single θ and do not care about the formula derivation, the Double Angle Calculator skips the derivation step and gives you 2θ, sin(2θ), cos(2θ), and tan(2θ) directly.

The solver reads θ and the unit selector, converts θ to a single radian representation, evaluates the three identities at 2θ, and adds a numeric identity check by computing 2 sin(θ) cos(θ) separately. Each output is the value the corresponding identity predicts, so the page is doing the same algebra a student would do by hand, only numerically.

• 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 formula output below is evaluated at this angle.
• sin(2θ), cos(2θ), tan(2θ): The three trig values at 2θ, each labeled 'by formula' so the output is unambiguous about which identity produced it.

The identity check on the results panel is a numeric verification of the sine formula. When the two numbers match to four decimal places, the rest of the formula outputs are trustworthy for the same input.

According to Wolfram MathWorld: Double-Angle Formulas, the three 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 page easy to follow. They are the same building blocks the identities are derived from, so once they are clear the formulas themselves fall out as one-line applications.

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

The tangent branch of the formula 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 solver takes a few seconds. Pick the unit, type the angle, and read the doubled angle plus all three formula outputs 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 solver 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 for the formulas.
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 the three formula outputs: sin(2θ), cos(2θ), and tan(2θ) appear in the same order as the formulas, 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 lists all three double angle identities, applies them to one input angle, and runs a numeric identity check saves time on homework, teaching prep, and quick sanity checks during longer derivations.

• • All three identities in one view: sin(2θ), cos(2θ), and tan(2θ) appear together with the formula box above them, 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 sine identity is visible, not just assumed.
• • Undefined tangent is handled cleanly: The solver reports 'undefined' for tan(2θ) when cos(2θ) is essentially zero, instead of an overflow number from the formula denominator.
• • Negative and out-of-range angles work: Inputs outside 0 to 360 and negative angles evaluate correctly because the solver uses Math.sin, Math.cos, Math.tan directly.

The biggest practical win is keeping the original angle, the doubled angle, the three formula outputs, and the identity check on the same screen.

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 solver returns the expected formula values, but a handful of factors control what the tool can and cannot do. Knowing them up front prevents the most common mistakes.

• • 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.
• • Floating-point arithmetic means the identity check matches sin(2θ) only to roughly 15 significant digits, so treat the check as a sanity check rather than an exact equality test for large angles.

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

According to Wikipedia: List of trigonometric identities, the three identities 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 formulas produce.