An arc of a circle with the same length as the radius of that circle subtends an angle of precisely 1 radian. The circumference subtends an angle of 2π radians.
Taken from the Wikipedia page on Radians.
Unit circle animation
The angle θ is filled in grey and its value in radians indicated next to the black dot.
The circle above has radius=1, so the horizontal and vertical coordinates of the black point correspond to cos(θ) and sin(θ).
cos(θ) is measured (in blue) along the horizontal axis and varies from -1 to 1.
sin(θ) is measured (in red) along the vertical axis and varies from -1 to 1.
Here is a similar diagram, but also including the tangent, which is given by the yellow line. The value of the tangent is the y-coordinate of the point at which an infinite line that extends from the origin to the point on the unit circle defined by the angle intersects the vertical line tangent to the circle at x=1.
The Unit Circle and the Graphs of Trigonometric Functions
The cosine graph generated from the unit circle.