Cos 2 Theta Calculator - Double Angle in Degrees or Radians

A cos 2 theta calculator applies the cosine double angle identity to any angle you enter and returns the cosine, sine, and tangent of the doubled angle 2θ from one base angle theta. The substitution, the Pythagorean step, and the unit conversion all run in one pass.

• • Double angle homework and exams: Solve problems that ask for cos(2θ), sin(2θ), or tan(2θ) from a known theta.
• • Physics and engineering formulas: Plug a measured theta into formulas with cos(2θ) or sin(2θ), like wave interference and AC power factor terms.
• • Reference value spot-checks: Verify textbook values like cos(60°) = 0.5 from theta = 30° or cos(120°) = −0.5 from theta = 60° while you study.

The identity that drives the tool is the cosine double angle formula, written in three equivalent ways: cos(2θ) = cos²(θ) − sin²(θ), cos(2θ) = 1 − 2 sin²(θ), and cos(2θ) = 2 cos²(θ) − 1. All three are mathematically the same and follow from setting a = b = θ in the cosine addition formula and applying the Pythagorean identity.

When you already have the doubled angle 2θ and need the cosine of it, the Cos Calculator runs the same cosine math in the forward direction and returns cos, sin, and the unit-circle point from a single angle input.

The tool reads the base angle theta and the chosen angle unit, converts theta to a single radian value, and then evaluates the cosine, sine, and tangent of the doubled angle 2θ directly. The same 2θ value is also written in degrees and as a multiple of pi so the result can drop straight into the rest of the problem.

• theta: The base angle in degrees, radians, or as a multiple of pi (for example 0.25 for pi/4).
• cos(2θ): The cosine of the doubled angle, in [-1, 1].
• tan(2θ): The tangent of the doubled angle, undefined when 2θ hits a vertical asymptote (90° + 180°k).

The calculator uses cos(2θ) = cos²(θ) − sin²(θ) directly. In degrees mode it multiplies theta by pi/180, in radians mode theta is used as-is, and in piX mode theta is multiplied by pi.

According to Wikipedia: Double-angle formulas, the cosine double angle identity can be written as cos(2θ) = cos²(θ) − sin²(θ), which is also equal to 1 − 2 sin²(θ) and 2 cos²(θ) − 1 by the Pythagorean identity.

When you only know cos(2θ) and need to recover the angle 2θ that produced it, the Cos 1 Calculator inverts the cosine for any value in [-1, 1] and gives the principal angle in degrees, radians, and pi form.

These four concepts are the building blocks for the cos 2 theta identity. Knowing them up front makes the worked examples and the limits of the formula much easier to read.

According to Wikipedia: Trigonometric functions, the cosine of any real angle is bounded between -1 and 1, so cos(2θ) inherits the same closed interval as cos(θ) regardless of how theta is entered.

If you want to double-check the relationship between cos(2θ) and 2θ, the Arccos Calculator reports the same principal arccos angle in degrees, radians, and pi form, which lines up directly with the doubled angle in the result panel.

Pick a unit, enter theta, and read cos(2θ), sin(2θ), tan(2θ), and 2θ from the result panel.

1. 1 Pick the angle unit: Choose degrees, radians, or pi times radians. The piX mode is convenient for reference angles like 0.25 (pi/4) or 0.5 (pi/2).
2. 2 Enter the base angle theta: Type the base angle into the input box. The calculator accepts any real number, including negative angles and angles larger than one full turn.
3. 3 Read cos(2θ) and the trig set: The primary panel shows cos(2θ), and the supporting rows show sin(2θ), tan(2θ), 2θ in degrees, radians, and as a multiple of pi, plus the original cos(θ) and sin(θ).
4. 4 Watch for the tan(2θ) undefined case: If theta = 45° + 90°k, tan(2θ) is shown as undefined with a short note. The other results are still reported normally.

For theta = 60° in degrees mode, the result panel shows cos(2θ) = -0.5, sin(2θ) ≈ 0.8660, tan(2θ) ≈ -1.7321, 2θ = 120° (0.6667 pi), cos(θ) = 0.5, and sin(θ) ≈ 0.8660. The same calculation in radians mode would use theta = pi/3 (1.0472) and produce the identical result.

When the rest of the problem uses a unit that the calculator does not display (for example gradians or turns), the Angle Converter is the fastest way to move 2θ between degrees, radians, pi multiples, and other angle scales.

A double angle tool that returns the full trig set and the doubled angle in three units covers the common cos 2 theta needs in one step.

• • Full double angle trig set at once: See cos(2θ), sin(2θ), and tan(2θ) from one input angle.
• • Three unit representations for 2θ: Read the doubled angle in degrees, radians, and as a multiple of pi without redoing the conversion.
• • Three input units, one calculation: Enter theta in degrees, radians, or pi multiples. The tool converts to a single radian value internally, so the same problem can be checked in whichever unit the surrounding work uses.
• • Original cos(θ) and sin(θ) read out: The panel also shows the original theta values so you can pick the identity form that matches your data.

All three identity forms agree on the same number, and the calculator keeps them aligned by computing cos(2θ) directly from the doubled angle rather than from whichever form you happened to pick. The form only changes when the data you already have changes.

When the result panel shows 2θ in radians and the rest of the geometry is in degrees, the Radians to Degrees Calculator handles that conversion in both directions so you can drop the doubled angle straight into the next step of the problem.

A few factors control what this tool can give you. Knowing them up front prevents the most common mistakes, especially when the doubled angle lands on a tangent asymptote or when theta is large.

• • The calculator works in real-valued cos 2 theta math. It does not compute complex-valued cos(2θ) for imaginary theta, which is rarely what classroom or applied problems need.
• • Floating-point arithmetic means cos(2θ), sin(2θ), and 2θ are only exact to roughly 15 significant digits. Round to the precision of the rest of the problem.

The sin(2θ) value in the result panel is the matching sine of the doubled angle, and the 2θ in degrees is the angle you would feed into a forward cosine tool to recover cos(2θ) from the same starting point. That round-trip is a quick way to spot a unit mix-up in the original problem.

According to Wolfram MathWorld: Double-Angle Formulas, the cos 2 theta identity follows from the cosine addition formula cos(a + b) = cos a cos b − sin a sin b by setting a = b = theta and applying sin²(θ) + cos²(θ) = 1.

When theta comes from a real right triangle, the Right Triangle Calculator lets you cross-check the cos 2 theta result against the other sides and the remaining angles of the triangle, which is the most reliable way to confirm the unit and the identity form.