Library / Foundations: Maths / FMA-60
Foundations: Maths
Cosine of a sum of angles
Cosine of two angles added together, in terms of the sines and cosines of each.
- Ref
- FMA-60
- Rev
- v1
- Field
- Foundations: Maths
- Result
- cos(A + B)
The compound-angle identity gives cos(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.
Cosine of two angles added together, in terms of the sines and cosines of each. Entering a negative B gives cos(A − B).
Calculator
- cos A cos B0.6124
- sin A sin B0.3536
- Direct evaluation, cos(A + B)0.2588
- 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 cos(75°) = 0.25882, matching the direct value. For 30° and 45° this is cos 75°, about 0.2588.
- Note the minus sign in the formula for cos(A + B): the cosine of a sum is smaller than the product of the cosines, whereas for cos(A − B) the sign is plus.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| A First angle | −180 to 180 ° | −0.9986 | 0.9986 | 1.997 | +7.2 ° | −47.6 % |
| B Second angle (negative for a difference) | −180 to 180 ° | −0.9945 | 0.9945 | 1.989 | +7.2 ° | −47.6 % |
Sweep of First angle
| First angle (°) | Cosine of the sum |
|---|---|
| −180 | −0.7071 |
| −135 | 0 |
| −90 | 0.7071 |
| −45 | 1 |
| 0 | 0.7071 |
| 45 | 2.22e-16 |
| 90 | −0.7071 |
| 135 | −1 |
| 180 | −0.7071 |
Sweep one variable
Variables and how each one contributes
Cosine 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 cosine of the sum falls as A increases while A + B is between 0° and 180°. Raising it by 5° (to 35°) moves the result by −32.9 %, from 0.2588 to 0.1736.
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 −32.9 %, from 0.2588 to 0.1736.
Worked example with the values above
- First product: cos A cos B = cos 30° × cos 45° = 0.61237.
- Second product: sin A sin B = 0.35355.
- Subtract: cos(A + B) = 0.61237 − 0.35355 = 0.25882.
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 cos(A + B) = cos A + cos B, which is false.
- Using a plus sign between the two products for a sum of angles (it is a minus).
Statistics and uncertainty
The identity is exact. Check results by evaluating the cosine of the summed angle directly, as the derived rows do.
Where it is used
- Resolving a rotated vector and combining two phase angles.
- Simplifying products of trigonometric functions in vibration and wave problems.
Maths you need
- Sine and cosine: ratios in a right triangle
- Degrees and radians: 180° = π radians