# Sine

## Definition 1

$$\sin A = \dfrac{\text{opposite}}{\text{hypotenuse}}$$

## Definition 2

THEOREM    The sine of any angle is equal to the cosine of its complementary angle. Similarly, the cosine of any angle is equal to the sine of its complementary angle. \begin{align} \sin \theta = \cos (90^\circ - \theta) \\ \cos \theta = \sin (90^\circ - \theta) \end{align}

Proof available

THEOREM    The sine function is periodic, with period $$2\pi$$.

Proof unavailable

THEOREM

The Maclaurin series of sine is

$$x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \dfrac{x^7}{7!} + \dfrac{x^9}{9!} - \ldots$$

Proof available

THEOREM    $$\forall a,b \in \mathbb{R} : \sin(a + b) = \sin a \cos b + \cos a \sin b$$

Proof available

COROLLARY    $$\forall a,b \in \mathbb{R} : \sin(a - b) = \sin a \cos b - \cos a \sin b$$

Proof available