written in the form that corresponds, in geometrical contexts, to the radian measurement of angles. The trigonometric functions also have simple and elegant series expansions when radians are used. For example, when x is in radians, the Taylor series for sin x becomes: If x were expressed in degrees, then the series would contain messy factors involving powers of /180: if x is the number of degrees, the number of radians is , so In a similar spirit, mathematically important relationships between the sine and cosine functions and the exponential function (see, for example, Euler's formula) can be elegantly stated, when the functions' arguments are in radians (and messy otherwise). Physics The radian is widely used in physics when angular measurements are required. For example, angular velocity is typically expressed in the unit radian per second (rad/s). One revolution per second corresponds to 2 radians per second. Similarly, the unit used for angular acceleration is often radian per second per second (rad/s2). For the purpose of dimensional analysis, the units of angular velocity and angular acceleration are s−1 and s−2 respectively. Likewise, the phase difference of two waves can also be expressed using the radian as the unit. For example, if the phase difference of two waves is (n⋅2) radians with n is an integer, they are considered to be in phase, whilst if the phase difference of two waves is () with n an integer, they are considered to be in antiphase. Prefixes and variants Metric prefixes for submultiples are used with radians. A milliradian (mrad) is a thousandth of a radian (0.001 rad), i.e. . There are 2 × 1000 milliradians (≈ 6283.185 mrad) in a circle. So a milliradian is just under of the angle subtended by a full circle. This unit of angular measurement of a circle is in common use by telescopic sight manufacturers using (stadiametric) rangefinding in reticles. The divergence of laser beams is also usually measured in milliradians. The angular mil is an approximation of the milliradian used by NATO and other military organizations in gunnery and targeting. Each angular mil represents of a circle and is % or 1.875% smaller than the milliradian. For the small angles typically found in targeting work, the convenience of using the number 6400 in calculation outweighs the small mathematical errors it introduces. In the past, other gunnery systems have used different approximations to ; for example