Contents

# Contents

## Idea

The sine function $\sin$ is one of the basic trigonometric functions.

It may be thought of as assigning to any angle the distance to a chosen axis of the point on the unit circle with that angle to that axis.

## Definitions

The sine function is the function $\sin \;\colon\; \mathbb{R} \to \mathbb{R}$ from the real numbers to themselves which is characterized by the following equivalent conditions:

1. $\sin$ is the unique solution among smooth functions to the differential equation/initial value problem

$sin'' = -sin$

(where a prime indicates the derivative) subject to the initial conditions

\begin{aligned} sin(0) &= 0 \\ sin'(0) & = 1 \,. \end{aligned}
2. (Euler's formula) $\sin$ is the imaginary part of the exponential function with imaginary argument

\begin{aligned} \sin(x) & = Im\left( \exp(i x) \right) \\ & = \frac{1}{2 i}\left( \exp(i x) - \exp(- i x)\right) \end{aligned} \,.
3. $\sin$ is the unique function which

1. is continuous at $0$

2. satisfies the inequality

$1 - x^2 \;\leq\; \frac{\sin(x)}{x} \;\leq\; 1$

and the equation

$\sin(3 x) \;=\; 3 \sin(x) - 4 (\sin(x))^3$

(see Trimble: Characterization of sine. Some additional discussion at the nForum.)

## Properties

### Relation to other functions

See at trigonometric identity.

### Roots

The roots of the sine function, hence the argument where its value is zero, are the integer multiples of pi $\pi \in \mathbb{R}$.