The sine function

The sine of a real number $t$ is given by the $y$−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]#: 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, 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 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