Library / Foundations: Maths / FMA-59
Foundations: Maths
Sine of a sum of angles
Sine of two angles added together, in terms of the sines and cosines of each.
- Ref
- FMA-59
- Rev
- v1
- Field
- Foundations: Maths
- Result
- sin(A + B)
The compound-angle identity gives sin(A + B) from the sines and cosines of the two angles. It holds for any angles; entering a negative B gives the identity for A − B, because sin(−B) = −sin B and cos(−B) = cos B. Angles are in degrees here and converted to radians internally.
Sine of two angles added together, in terms of the sines and cosines of each. Entering a negative B gives sin(A − B).
Calculator
- sin A cos B0.3536
- cos A sin B0.6124
- Direct evaluation, sin(A + B)0.9659
- Sum of the angles A + B75 °
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.
- For 30° and 45° the identity gives sin(75°) = 0.96593, which equals the direct evaluation of the sine of 75°. For 30° and 45° this is the value of sin 75°, about 0.9659.
- Setting B = 90° − A shows the identity reduces to sin²A + cos²A = 1, a useful check.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| B Second angle (negative for a difference) | −180 to 180 ° | −1 | 1 | 2 | +7.2 ° | +2.57 % |
| A First angle | −180 to 180 ° | −0.9986 | 0.9986 | 1.997 | +7.2 ° | +2.57 % |
Sweep of First angle
| First angle (°) | Sine of the sum |
|---|---|
| −180 | −0.7071 |
| −135 | −1 |
| −90 | −0.7071 |
| −45 | 0 |
| 0 | 0.7071 |
| 45 | 1 |
| 90 | 0.7071 |
| 135 | 2.22e-16 |
| 180 | −0.7071 |
Sweep one variable
Variables and how each one contributes
Sine of the sum
What the equation gives you. Everything else feeds it.
First angle
The first angle, in degrees. Changes the result through cos A and sin A; the sine of the sum rises as A increases while A + B is below 90°, then falls. Raising it by 5° (to 35°) moves the result by +1.95 %, from 0.9659 to 0.9848.
Second angle (negative for a difference)
The second angle, in degrees. Enter a negative value to obtain the identity for A − B. Changes the result in the same way as A, because the two angles enter symmetrically. Raising it by 5° (to 50°) moves the result by +1.95 %, from 0.9659 to 0.9848.
Worked example with the values above
- First product: sin A cos B = sin 30° × cos 45° = 0.5 × 0.70711 = 0.35355.
- Second product: cos A sin B = 0.86603 × 0.70711 = 0.61237.
- Add: sin(A + B) = 0.96593.
Assumptions and limits
- Angles in degrees on the page (radians in a spreadsheet's SIN and COS).
- Any angles, positive or negative.
Common mistakes
- Writing sin(A + B) = sin A + sin B, which is false.
- Giving angles in degrees to a function that expects radians.
Statistics and uncertainty
The identity is exact. Used numerically, keep the angles in consistent units; a degrees-for-radians slip is by far the commonest source of error.
Where it is used
- Resolving a rotated vector and combining two phase angles.
- Deriving wave, rotation and transformation formulas.
Maths you need
- Sine and cosine: ratios in a right triangle
- Degrees and radians: 180° = π radians