What this calculator does
The double angle identity for cosine gives cos(2θ) directly from θ, without needing to double the angle and look up its cosine separately. The most common form is cos(2θ) = cos²(θ) minus sin²(θ), and it turns up throughout trigonometry, physics and engineering wherever an angle is being doubled, such as in oscillation and wave problems.
The identity has three equivalent forms, all producing the same result: cos²(θ) − sin²(θ), 2cos²(θ) − 1, and 1 − 2sin²(θ). They are algebraically identical because sin²(θ) + cos²(θ) = 1, so any one of them can be substituted for another depending on which is more convenient for the problem at hand. This calculator shows all three so the working can be checked.
The formula
Convert the angle to radians if it was entered in degrees, then compute cos(θ) and sin(θ) directly. cos(2θ) is cos(θ) squared minus sin(θ) squared; the calculator also shows the same answer from the other two equivalent forms of the cos2theta formula as a cross-check.
| Term | Meaning |
|---|---|
| θ (theta) | The angle being doubled, entered in degrees or radians. |
| cos(2θ) | The cosine of double the entered angle, found via the double angle identity rather than by doubling θ and taking its cosine from a table. |
| Double angle identity | An identity expressing a trig function of 2θ purely in terms of trig functions of θ. |
The inputs explained
| Field | What to enter |
|---|---|
| Angle (θ) (degrees) | The angle θ, in whichever unit is selected below. |
| Angle unit | Whether the angle above is in degrees or radians. |
When to use it
Checking a cos 2 theta formula by hand
Working through a trigonometry problem that needs cos(2θ) can be checked here to catch a sign error or an arithmetic slip before it propagates through the rest of the working.
Physics and engineering problems
Oscillation, wave interference and AC power calculations often involve a doubled angle inside a cosine term; this gives that value directly from the base angle.
Studying the identity itself
Comparing the three equivalent forms for the same angle shows why they always agree, which is a useful check when first learning the double angle identities.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What is cos(2θ) for common angles?
cos(2θ) across a spread of common angles, from 0° to 90°.
| θ | cos(2θ) | cos(θ) |
|---|---|---|
| 0° | 1.000 | 1.000 |
| 15° | 0.8660 | 0.9659 |
| 30° | 0.5000 | 0.8660 |
| 45° | 2.2204e-16 | 0.7071 |
| 60° | -0.5000 | 0.5000 |
| 75° | -0.8660 | 0.2588 |
| 90° | -1.000 | 6.1232e-17 |
Questions
What is the cos2theta formula?
cos(2θ) = cos²(θ) − sin²(θ). It is equivalent to 2cos²(θ) − 1 and to 1 − 2sin²(θ); all three give the same result because sin²(θ) + cos²(θ) = 1.
What is the cos 2 theta formula used for?
It lets you find the cosine of a doubled angle directly from the original angle, without separately computing 2θ and looking up its cosine. It appears often in calculus, physics and engineering when simplifying expressions involving squared sine or cosine terms.
Is cos(2θ) the same as 2cos(θ)?
No, and this is a common mix-up. cos(2θ) uses the double angle identity and is not simply double the value of cos(θ). For example, at θ = 30°, cos(θ) ≈ 0.866, but cos(2θ) = cos(60°) = 0.5, not 1.732.
Does the formula work in radians as well as degrees?
Yes. The identity itself is unit-independent; just make sure the angle is converted to radians before it goes into the cosine and sine functions, which this calculator handles automatically based on the unit selected.
For the law of cosines in triangle problems, see the triangle trigonometry calculator. For general angle unit conversion, see the angle conversion calculator.