Library / Geometry, coordinates, surveying and trigonometry / GEO-45
Geometry, coordinates, surveying and trigonometry
Single harmonic equivalent of a sine and cosine sum
Amplitude R of the single sine wave R sin(x + φ) that equals a sin x + b cos x.
- Ref
- GEO-45
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- R
A sum a sin x + b cos x is a single sine wave R sin(x + φ), with R = √(a² + b²) and φ = atan2(b, a). It follows from expanding R sin(x + φ) = R sin x cos φ + R cos x sin φ and matching coefficients. This amplitude is shown here; the phase is in the derived rows. If a = b = 0, R = 0 and the phase is not defined.
Amplitude R of the single sine wave R sin(x + φ) that equals a sin x + b cos x. The phase φ = atan2(b, a) is in the derived rows. If a = b = 0 the amplitude is zero and the phase is not defined.
Calculator
- Phase φ = atan2(b, a)0.9273 rad
- Phase in degrees53.13 °
- Value at x = 0, which equals b4
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.
- 3 sin x + 4 cos x = 5 sin(x + 0.9273), a wave of amplitude 5 shifted 53.13°.
- This is how two vibrations or two AC voltages of the same frequency combine: the amplitudes add like the sides of a right triangle, not directly.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| b Cosine coefficient | −10 to 10 | 3 | 10.44 | 7.44 | +0.4 | +6.51 % |
| a Sine coefficient | −10 to 10 | 4 | 10.77 | 6.77 | +0.4 | +5 % |
Sweep of Sine coefficient
| Sine coefficient | Amplitude |
|---|---|
| −10 | 10.77 |
| −7.5 | 8.5 |
| −5 | 6.403 |
| −2.5 | 4.717 |
| 0 | 4 |
| 2.5 | 4.717 |
| 5 | 6.403 |
| 7.5 | 8.5 |
| 10 | 10.77 |
Sweep one variable
Variables and how each one contributes
Amplitude
What the equation gives you. Everything else feeds it.
Sine coefficient
The coefficient of sin x, a plain number (or in any unit shared with b). Raises R, mostly when a is large compared with b. Raising it by 0.5 (to 3.5) moves the result by +6.3 %, from 5 to 5.315.
Cosine coefficient
The coefficient of cos x, in the same unit as a. Raises R, mostly when b is large compared with a. Raising it by 0.5 (to 4.5) moves the result by +8.17 %, from 5 to 5.408.
Worked example with the values above
- Squares: a² + b² = 9 + 16 = 25.
- Phase: φ = atan2(4, 3) = 0.9273 rad.
- R = √25 = 5.
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 atan(b/a) in place of atan2(b, a), which puts the phase in the wrong quadrant when a is negative.
- Adding the amplitudes a + b instead of combining them as a root of the sum of squares.
- Mixing a sine form with a phase and a cosine form with a phase.
Statistics and uncertainty
The amplitude is a root of a sum of squares, so errors in the larger coefficient dominate, and the phase is poorly determined when both coefficients are small.
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