The cosine function
The
cosine of a real number $t$ is given by the $x$-coordinate of the point $P$ in the following diagram, in which $t$ is the length of the arc shown.
Graph of the conside function:
As you might expect, the graph of $y = \cos(t)$ has the same cyclical shape as that of $y = \sin(t)$. The only difference between the two is a "phase shift" (see the figure).
y = cos(t)
#[The graph of $y = \sin(t)$ is shown in a lighter shade.][La gráfica de $y = \sin(t)$ se muestra en un tono más claro.]#
#[Notes][Notas]#
- The graph of $\cos(t)$ is symmetric about the $y$-axis; replacing $t$ by $-t$ gives the same value for $\cos(t)$:
$\displaystyle \cos(-t) = \cos(t)$.
By comparison, we saw that the graph of $\sin(t)$ is antisymmetric about the $y$-axis; replacing $t$ by $-t$ results in a sign change:
$\displaystyle \sin(-t) = -\sin(t)$.
-
The graph of the sine can be obtained by shifting the graph of cosine to the right by $\pi/2$, so
$\displaystyle \sin(t) = \cos\left(t-\frac{\pi}{2}\right)$.
#[Examples][Ejemplos]#
When $t = 0$ or $2\pi$, the point $P$ is on the $x$-axis with $x$-coordinate $1$, so that
$\cos(0) = \cos(2\pi) = 1$ \gap[20] \t
When the point $P$ has moved a distance of $\dfrac{\pi}{2}$ in either direction and so makes an angle of 90° with the positive $x$-axis, its $x$-coordinate is 0, so
$\displaystyle \cos\left(\pm\frac{\pi}{2}\right)= 0$ \gap[20] \t
