Trigonify

Topics · 3 identities

Sum & Difference

Expanding sine, cosine, and tangent of a sum or difference of angles.

Sine sum and difference formula: sin(α ± β) = sin α cos β ± cos α sin β

sin⁡(α±β)=sin⁡αcos⁡β±cos⁡αsin⁡β\sin(\alpha \pm \beta) = \sin\alpha \cos\beta \pm \cos\alpha \sin\beta

sin(α ± β) = sin α cos β ± cos α sin β. The signs match: a plus inside gives a plus outside. Use it to get exact values for angles you can build from known ones - sin 75° = sin(45° + 30°) - and to derive the double-angle formula by setting β = α.

Open identity

Cosine sum and difference formula: cos(α ± β) = cos α cos β ∓ sin α sin β

cos⁡(α±β)=cos⁡αcos⁡β∓sin⁡αsin⁡β\cos(\alpha \pm \beta) = \cos\alpha \cos\beta \mp \sin\alpha \sin\beta

cos(α ± β) = cos α cos β ∓ sin α sin β. The signs are opposite: a plus inside gives a minus outside. That sign flip is the most commonly dropped detail in the whole topic, and it comes from cosine being even while sine is odd.

Open identity

Tangent sum and difference formula: tan(α ± β)

tan⁡(α±β)=tan⁡α±tan⁡β1∓tan⁡αtan⁡β\tan(\alpha \pm \beta) = \dfrac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha \tan\beta}

tan(α ± β) = (tan α ± tan β) / (1 ∓ tan α tan β). The numerator takes the same sign as the angle sum and the denominator takes the opposite one. It is the formula behind the angle-between-two-lines result, where the tangents are the two slopes.

Open identity