How this calculator works
Sum identities combine the function values at two separate angles. Changing the sign of the second angle gives the difference identities. Reciprocal functions can be evaluated from the sine or cosine of the resulting angle. A reverse calculation selects a principal target angle; it does not enumerate every periodic solution for β. Singular ratios are shown as Undefined.
Formula
sin(α±β)=sin α cos β±cos α sin β; cos(α±β)=cos α cos β∓sin α sin β; tan(α±β)=(tan α±tan β)/(1∓tan α tan β) where defined.
Worked example
For α = 45° and β = 30°, sin 75° = (√6+√2)/4 ≈ 0.9659258263 and sin 15° = (√6-√2)/4 ≈ 0.2588190451.
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Frequently asked questions
Can I add the sine values directly?
No. sin(α+β) generally differs from sin α + sin β. The cross-products account for both angles.
What if tan α is undefined but tan(α+β) exists?
The tool evaluates the resulting angle directly, so it can return a valid value even when an intermediate tangent identity would be inconvenient.
Can reverse mode give a negative β?
Yes. A negative angle may be needed to reach the chosen principal target angle from α.