The addition and subtraction formulae

The functions $\sin$ and $\cos$ are not linear: in general $\sin(\alpha+\beta)\ne\sin\alpha+\sin\beta$. We therefore need a dedicated rule for the sine and cosine of a sum or a difference of angles. These are the addition and subtraction formulae, valid for any pair of angles.

Property — Addition and subtraction formulae

For every $\alpha,\beta\in\mathbb{R}$:

\cos(\alpha-\beta) &= \cos\alpha\cos\beta + \sin\alpha\sin\beta \\ \cos(\alpha+\beta) &= \cos\alpha\cos\beta - \sin\alpha\sin\beta \\ \sin(\alpha+\beta) &= \sin\alpha\cos\beta + \cos\alpha\sin\beta \\ \sin(\alpha-\beta) &= \sin\alpha\cos\beta - \cos\alpha\sin\beta \\ \tan(\alpha+\beta) &= \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta} \\ \tan(\alpha-\beta) &= \frac{\tan\alpha - \tan\beta}{1 + \tan\alpha\tan\beta} \end{aligned}$$

Notice the pattern of signs: in the cosine the sign is reversed (sum of angles, difference of products), whereas in the sine the sign is preserved. These six formulae are the foundation on which all the others in this chapter rest: from the duplication formulae to the bisection ones.

Links

Topics: Trigonometric formulae
Concepts: Addition formulae
Functions: Cosine · Sine · Tangent
Methods: Addition formulae
Skills: Using formulae