Formulas & tables/Mathematics/Trigonometry/Sine of a sum or difference of angles

Sine of a sum or difference of angles

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

Result unit: — (liczba)

What it describes

The sine of a sum or difference of two angles.

Symbols

α,β\alpha,\beta
angles

Where it comes from

The sine of a sum or difference of angles — it lets you compute, for example, sin⁡75∘\sin75^\circ as sin⁡(45∘+30∘)\sin(45^\circ+30^\circ). Watch the order of the signs.

When to use it

Non-standard angles, identities, proofs.

Example

Compute sin⁡75∘=sin⁡(45∘+30∘)\sin75^\circ=\sin(45^\circ+30^\circ).
Substitutionsin⁡45cos⁡30+cos⁡45sin⁡30\sin45\cos30+\cos45\sin30.
Result=6+24≈0.966=\dfrac{\sqrt6+\sqrt2}{4}\approx0.966.

Related formulas

Cosine of a sum or difference of angles Sine of a double angle

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