Home

Sum and Difference Formulas Calculator

Online angle addition calculator: enter α and β to find sin(α+β) and cos(α+β); the core identities from which difference and double-angle formulas follow.

展开更多 ▾
sin(α+β)=sinαcosβ+cosαsinβ
In degrees
In degrees

📖 Tutorial | Addition

1. Definition
The sum (and difference) expressed via the two angles: sin(α+β)=sinαcosβ+cosαsinβ, cos(α+β)=cosαcosβ−sinαsinβ.
2. Formula & Notation
α, βTwo angles
sin(α+β)Sine of the sum
3. Properties & Laws
  • sin(α+β)=sinαcosβ+cosαsinβ;
  • cos(α+β)=cosαcosβ−sinαsinβ;
  • Replace β with −β for the difference;
  • tan(α+β)=(tanα+tanβ)/(1−tanαtanβ).
4. Steps
  1. Enter α, β (degrees);
  2. Click "Calculate";
  3. Or click "Load Example".
5. Worked Examples
Example: sin75°=sin30cos45+cos30sin45≈0.9659.
6. Common Pitfalls
It is cross-multiplied terms, not sinα+sinβ;
The cosine formula has a minus;
Keep angle units consistent.

❓ FAQ | Addition

Sine sum?
sinαcosβ+cosαsinβ.
Cosine sum?
cosαcosβ−sinαsinβ.
Difference?
Replace β with −β.
Tangent sum?
(tanα+tanβ)/(1−tanαtanβ).
Equal to sinα+sinβ?
No.
Check?
sin(30+30)=sin60.
Uses?
Non-special angles, simplifying, proving.