Hi,
The sine of 10 degrees is about 0.1736 so twice that is about 0.347 or roughly 0.35, so if you hold the LED about 10 inches away from a flat object and the LED has a circular pattern then the circle you see lit will be roughly 0.35 times 10 or 3.5 inches in diameter. This is just approximate though but gives you some idea what to expect.
Sometimes there is also what they call "splash light" which is a little more light that shoots off to the sides but is much less bright.
The tight angle white LED's are used for various lighting purposes like flashlights. The fact that they have such a tight angle means sometimes we dont need a focusing lens.
Many of the high power LEDs have a much wider beam pattern and require some sort of focusing mechanism to get most of the light directed forward in a narrow beam. They make a lot of different lenses for this purpose that match with the model of the LED so that it fits right over the plastic of the LED. They typically take a 120 degree beam and turn it into a 20 to 40 degree beam with relatively good efficiency.