Library / Geometry, coordinates, surveying and trigonometry / GEO-42
Geometry, coordinates, surveying and trigonometry
Double-angle cosine from a sine value
Cosine of twice an angle from the sine of the angle, s = sin θ, which must lie between −1 and 1.
- Ref
- GEO-42
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- \cos 2θ
The double-angle formula for cosine has three forms: cos²θ − sin²θ, 2cos²θ − 1 and 1 − 2sin²θ. The last one needs only s = sin θ, and not the quadrant of θ, so it is useful when a question gives only the sine of an angle. It must satisfy −1 ≤ s ≤ 1.
Cosine of twice an angle from the sine of the angle, s = sin θ, which must lie between −1 and 1. It does not need the quadrant of θ.
Calculator
- cos θ magnitude, √(1 − s²)0.8
- sin 2θ magnitude, 2|s|√(1 − s²)0.96
- Check, cos²2θ + sin²2θ1
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.
- With sin θ = 0.6, cos 2θ = 0.28, whatever quadrant θ is in.
- At sin θ = 1 (θ = 90°) the result is −1, and at sin θ = 0 it is 1, the two limits of the formula.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| s Sine of the angle, sin θ | −1 to 1 | −1 | 1 | 2 | +0.04 | −35.4 % |
Sweep of Sine of the angle, sin θ
| Sine of the angle, sin θ | Cosine of the double angle |
|---|---|
| −1 | −1 |
| −0.75 | −0.125 |
| −0.5 | 0.5 |
| −0.25 | 0.875 |
| 0 | 1 |
| 0.25 | 0.875 |
| 0.5 | 0.5 |
| 0.75 | −0.125 |
| 1 | −1 |
Sweep one variable
Variables and how each one contributes
Cosine of the double angle
What the equation gives you. Everything else feeds it.
Sine of the angle, sin θ
The sine of the angle, a plain number from −1 to 1. Lowers cos 2θ with the square of s, so the sign of s does not matter. Raising it by 0.05 (to 0.65) moves the result by −44.6 %, from 0.28 to 0.155.
Worked example with the values above
- Square the sine: s² = 0.6² = 0.36.
- Double it: 2s² = 0.72.
- cos 2θ = 1 − 0.72 = 0.28.
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