There are two possible definitions of the trigonometric ratios:
The trigonometric ratios can be defined for angles greater than $0^\circ$ and less than $90^\circ$ using right triangles. In particular, $\sin(\theta)$ is defined as the ratio of the lengths of the opposite leg and the hypotenuse, and $\cos(\theta)$ is defined as the ratio of the lengths of the adjacent leg and the hypotenuse.
The trigonometric ratios can be defined for any angle using the unit circle. In this definition, $\sin(\theta)$ is the $y$-coordinate of a point on the unit circle with angle $\theta$, and $\cos(\theta)$ is the $x$-coordinate of a point on the unit circle with angle $\theta$.
The unit circle definition is the same as the triangle definition for angles between $0$ and $90^\circ$, but is more general since it works for any angle. The following picture from Wikipedia illustrates this definition:
For each point, the $x$-coordinate is the cosine, and the $y$-coordinate is the sine.
This picture only shows angles between $0^\circ$ and $360^\circ$, but you can extend to less than $0^\circ$ by continuing clockwise around the circle, or to greater than $360^\circ$ by continuing counterclockwise.
The following pictures show graphs of $\sin(x)$ and $\cos(x)$ for $-2\pi\leq x\leq2\pi$. (The $x$-axis is the angle measured in radians.)