Library / Geometry, coordinates, surveying and trigonometry / GEO-43
Geometry, coordinates, surveying and trigonometry
Double-angle sine from a tangent value
Sine of twice an angle from the tangent u = tan θ.
- Ref
- GEO-43
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- \sin 2θ
The double-angle formula sin 2θ = 2 sin θ cos θ can be written in terms of u = tan θ. Dividing top and bottom of 2 sin θ cos θ / (sin²θ + cos²θ) by cos²θ gives 2u/(1 + u²). It works for every finite u, positive or negative. For extremely large u, 2/(u + 1/u) avoids overflow.
Sine of twice an angle from the tangent u = tan θ. It follows from 2 sin θ cos θ divided by sin²θ + cos²θ. For extremely large u use 2/(u + 1/u).
Calculator
- cos 2θ, (1 − u²)/(1 + u²)−0.6
- tan 2θ, 2u/(1 − u²)−1.333
- Angle θ, arctan u63.43 °
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- An angle with tangent 2 (63.43°) has a double angle with sine 0.8.
- The result is never above 1, and reaches 1 at u = 1 (45°), where the double angle is a right angle.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| u Tangent of the angle, tan θ | −10 to 10 | −0.96 | 0.96 | 1.92 | +0.4 | −11.2 % |
Sweep of Tangent of the angle, tan θ
| Tangent of the angle, tan θ | Sine of the double angle |
|---|---|
| −10 | −0.198 |
| −7.5 | −0.262 |
| −5 | −0.3846 |
| −2.5 | −0.6897 |
| 0 | 0 |
| 2.5 | 0.6897 |
| 5 | 0.3846 |
| 7.5 | 0.262 |
| 10 | 0.198 |
Sweep one variable
Variables and how each one contributes
Sine of the double angle
What the equation gives you. Everything else feeds it.
Tangent of the angle, tan θ
The tangent of the angle, a plain number. It can be any finite real value. Raises sin 2θ for small u and lowers it once u passes 1, where it peaks. Raising it by 0.25 (to 2.25) moves the result by −7.22 %, from 0.8 to 0.7423.
Worked example with the values above
- Denominator: 1 + u² = 1 + 4 = 5.
- Numerator: 2u = 4.
- sin 2θ = 4 / 5 = 0.8.
Assumptions and limits
- The inputs are the exact sine, cosine or tangent values of a real angle.
- Where a square root is taken, the stated range of the angle fixes the sign.
Common mistakes
- Using the formula when the tangent is undefined, such as at 90°.
- Taking the positive square root outside the range where the sign is positive.
- Mixing degrees and radians when using a calculator for the final angle.
Statistics and uncertainty
Identities are exact, so the only error is numerical. Tangent values near a pole (uv close to 1) magnify rounding, and a square root of a number near zero loses digits.
Where it is used
- Simplifying expressions and solving trigonometric equations in mathematics and physics.
- Combining two sinusoids of the same frequency into one wave, as in phasors, AC circuits and vibration.
Maths you need
- Sine, cosine and tangent: ratios from a right triangle or the unit circle
- Pythagorean identity: sin² + cos² = 1
- Phase: the horizontal shift of a wave