Quick answer
Double angle identities: sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ − sin²θ and tan 2θ = 2 tan θ ÷ (1 − tan²θ). For θ = 30°: sin 60° = 0.8660, cos 60° = 0.5 and tan 60° = 1.7321.
Enter an angle and its unit. The calculator returns sin, cos and tan of the double angle.
Related tools: Reference Angle Calculator: Acute angle between the terminal side and the x-axis. Coterminal Angle Calculator: Angles sharing the same terminal side. Angle of Elevation Calculator: Angle of elevation from a height difference and horizontal distance.
How to use the Double Angle Calculator
- Enter angle.
- Choose unit: Degrees and Radians.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Angle | A number | — |
| Unit | Choose one: Degrees; Radians | Starts at Degrees |
Formula and method
Formula
sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ
IEEE-754 double precision; shown to at most 6 decimal places.
Evaluates sin, cos and tan at 2θ.
Worked examples
This example is verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Angle = 30; Unit = Degrees.
| Result | Value |
|---|---|
| sin 2θ | 0.866025 |
| cos 2θ | 0.5 |
| tan 2θ | 1.732051 |
The first line is the headline result (sin 2θ). Below it the double Angle Calculator lists cos 2θ and tan 2θ, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Check
sin 30° = 0.5 and cos 30° = 0.8660, so 2 × 0.5 × 0.8660 = 0.8660 ✓. cos²30° − sin²30° = 0.75 − 0.25 = 0.5 ✓.
Where tan fails
tan 2θ is undefined where cos 2θ is zero (θ = 45°, for example). Alternative forms of cos 2θ are 2cos²θ − 1 and 1 − 2sin²θ.
How the result changes with angle
Holding every other input at the first worked example, this table changes only angle. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Angle | sin 2θ | cos 2θ | tan 2θ |
|---|---|---|---|
| 15 | 0.5 | 0.866025 | 0.57735 |
| 30 | 0.866025 | 0.5 | 1.732051 |
| 60 | 0.866025 | -0.5 | -1.732051 |
| 120 | -0.866025 | -0.5 | 1.732051 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Angle: left empty shows "Enter angle." beside that field and no result.
Choosing between the Double Angle Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Reference Angle Calculator](/math/reference-angle/) | angle | Reference angle |
| [Coterminal Angle Calculator](/math/coterminal-angle/) | angle | Smallest non-negative coterminal angle |
| [Angle of Elevation Calculator](/math/angle-of-elevation-calculator/) | height difference and horizontal distance | Angle of elevation |
| [Angle of Depression Calculator](/math/angle-of-depression-calculator/) | height difference and horizontal distance | Angle of depression |
| [Half Angle Calculator](/math/half-angle-calculator/) | angle and unit | sin(θ/2) |
Background: trigonometry
In a right triangle, sin θ is the opposite side over the hypotenuse, cos θ the adjacent side over the hypotenuse and tan θ the opposite over the adjacent. Their reciprocals are cosecant, secant and cotangent. At 30° the values are sin = 0.5, cos = 0.866025 and tan = 0.577350.
Check the angle unit before you calculate: degrees convert to radians by multiplying by π/180, so 30° is about 0.5236 rad. Some values are undefined, for example tan 90°, because the cosine is zero there and the ratio would divide by zero.
Assumptions and limitations
- tan undefined where cos is zero
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for angle to confirm this.
Frequently asked questions
What is sin 90° using double angle?
2 sin 45° cos 45° = 1.
Why is it useful?
It simplifies integrals and equations.
Does it work in radians?
Yes, choose the unit.
What does the Double Angle Calculator calculate?
Double-angle identities evaluated at an angle. It returns sin 2θ, cos 2θ and tan 2θ.
What formula does the Double Angle Calculator use?
sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ. Evaluates sin, cos and tan at 2θ.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. tan undefined where cos is zero.
What does the “Unit” option change?
It sets how the double Angle is worked out. The choices are Degrees and Radians.
What is an example of a double Angle result?
With angle 30 and unit Degrees, the sin 2θ is 0.866025.
Related calculators
- Reference Angle Calculator — Acute angle between the terminal side and the x-axis.
- Coterminal Angle Calculator — Angles sharing the same terminal side.
- Angle of Elevation Calculator — Angle of elevation from a height difference and horizontal distance.
- Angle of Depression Calculator — Angle of depression from a height difference and horizontal distance.
- Half Angle Calculator — Half-angle identities evaluated at an angle.
- Trigonometry Calculator — All six trigonometric functions of an angle.
Sources
- Standard trigonometric definitions on the unit circle (NIST DLMF §4.14)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.