**The sine function**
The

**sine** of a real number $t$ is given by the $y$âˆ’coordinate (height above the $x$-axis) of the point $P$ in the following diagram, in which $t$ is the length of the arc shown.

**#[Note][Nota]#:** We can also think of the quantity $t$ as measuring the angle $QOP$ it spans; the quantity $t$ it is then the

**radian measure** of the angle $Q0P$, and $\sin(t)$ is often called the

**sine of the angle $t$ radians.**
**#[Graph][Gráfica]#:**
*y* = #[sin][sen]#(*t*)
For instance, as the circumference of the entire circle is $2\pi$, when $t = 2\pi$, the point $P$ has moved around the entire circle, bringing it back tp its initial position on the $x$-axis at a height of $0$. Thus,

#[$\sin(2\pi) = 0$][$\text{sen}(2\pi) = 0$]# \gap[20] \t

Similarly, when the point $P$ has moved a quarter of the way around the circle, a distance of $\dfrac{2\pi}{4} = \dfrac{\pi}{2}$, it makes an angle of 90° with the positive $x$-axis and it is at its highest position, so

#[$\sin\left(\dfrac{\pi}{2}\right) = 1$][$\text{sen}\left(\dfrac{\pi}{2}\right) = 1$]# \gap[20] \t

A negative value of $t$ would correspond to moving the point $P$

*clockwise* through a distance of $|t|$. So, for instance, $t = -\dfrac{\pi}{2}$ would move it down to the right, quarter way around the circle to the lowest point, so

#[$\sin\left(-\dfrac{\pi}{2}\right) = -1$][$\text{sen}\left(-\dfrac{\pi}{2}\right) = -1$]# \gap[20] \t