Formulas & tables/Mathematics/Trigonometry/Sum and difference of sines

Sum and difference of sines

sin⁡α+sin⁡β=2sin⁡α+β2cos⁡α−β2\sin\alpha+\sin\beta=2\sin\tfrac{\alpha+\beta}{2}\cos\tfrac{\alpha-\beta}{2}

Result unit: — (liczba)

What it describes

Converting the sum of the sines of two angles into a product.

Symbols

α,β\alpha,\beta
angles

Where it comes from

It converts a sum of sines into a product — useful when solving equations and in the analysis of waves (beats).

When to use it

Trigonometric equations, simplifying sums, wave physics.

Example

Convert sin⁡3x+sin⁡x\sin 3x+\sin x into a product.
Substitutionα+β2=2x\tfrac{\alpha+\beta}{2}=2x, α−β2=x\tfrac{\alpha-\beta}{2}=x.
Result2sin⁡2xcos⁡x2\sin 2x\cos x.

Related formulas

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

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