Often, angles are measured in degrees. However, some problems in physics (and math) are simplified if angles are measured in radians. Radian measure involves the ratio of a part of a circumference of a circle to the radius of a circle.

Figure 1 - Definition of the radian

In Figure 1, the angle q is given in radians by s/r, where the arc length s is measured along the circumference of the circle. For example, if s=10 cm and r=20 cm, then q = 10cm/20cm = 0.5 (radians). If the arc length s equals the radius r (s=r) then the angle subtended by s is one (1) radian. (see Figure 2 where the angle shown is approximately one radian)

Figure 2 - The size of a radian

Note that "radian" is a ratio of lengths- the lengths need to be in the same units.

How many degrees is equivalent to one radian? Supposed the angle is 360^o. The arc length subtended by a 360^o angle is the circumference of the circle.

Since:

the measurement of the angle in radians is

That is, a 360^o angle is equivalent to 2p radians.

a 180^o angle is equivalent to p radians

Also, 1 radian = 57.3^o (approximately)

Many times in optics we are confronted with equations that are made much simpler by the use of the "small angle approximation". Figure 3 shows a "small" angle. You can see that s/r, the size of the angle in radians, is very close to y/x, where y is the vertical leg of a right triangle. Further, s/r is very close in value to y/r, where r is the radius of the circle as well as the hypotenuse of the right triangle.

Figure 3 - The red arc is a portion of a circle's circumference. Its radius is also shown in red (horizontal line) and in black (hypotenuse of a right triangle). Note that the horizontal leg of the right triangle is approximately the same length as the radius or the circle and the vertical leg of the right triangle is approximately the same length as the arc of the circle subtended by the angle theta

Putting these ideas together, along with the definitions of sine and tangent,

You can try this on your calculator (Be sure your calculator is in radian mode, not degree mode.) For example, if q = 0.05 radians, sin q = 0.049979 and tan q = 0.5004.

You will find that as the angle becomes larger (0.05 radians is equivalent to about 2.9 degrees) the sine and tangent and angle in radians begin to diverge in value. When is the small angle approximation valid? That depends on the needed accuracy of the calculation. Using the small angle approximation, Snell's law becomes n₁q₁ = n₂q₂. (The sines in the Snell's law equation have been replaced by the angles in radians.) This leads to the thin lens equation (we called it the paraxial approximation)- good enough for the lenses in the OSA optics kit, but not good enough for designing an expensive compound camera lens.