Small-angle approximation
For small angles, the trigonometric functions sine, cosine, and tangent can be calculated with reasonable accuracy by the following simple approximations: provided the angle is measured in radians.
Angles measured in degrees must first be converted to radians by multiplying them by
These approximations have a wide range of uses in branches of physics and engineering, including mechanics, electromagnetism, optics, cartography, astronomy, and computer science.
[1][2] One reason for this is that they can greatly simplify differential equations that do not need to be answered with absolute precision.
There are a number of ways to demonstrate the validity of the small-angle approximations.
The most direct method is to truncate the Maclaurin series for each of the trigonometric functions.
The red section on the right, d, is the difference between the lengths of the hypotenuse, H, and the adjacent side, A.
As is shown, H and A are almost the same length, meaning cos θ is close to 1 and θ2/2 helps trim the red away.
The opposite leg, O, is approximately equal to the length of the blue arc, s. Gathering facts from geometry, s = Aθ, from trigonometry, sin θ = O/H and tan θ = O/A, and from the picture, O ≈ s and H ≈ A leads to:
A more careful application of the squeeze theorem proves that
for small values of θ. Alternatively, we can use the double angle formula
The Taylor series expansions of trigonometric functions sine, cosine, and tangent near zero are:[5]
Thus for many purposes it suffices to drop the cubic and higher terms and approximate the sine and tangent of a small angle using the radian measure of the angle,
, and drop the quadratic term and approximate the cosine as
If additional precision is needed the quadratic and cubic terms can also be included,
By using the MacLaurin series of cosine and sine, one can show that
Furthermore, it is not hard to prove that the Pythagorean identity holds:
: for each order of magnitude smaller the angle is, the relative error of these approximations shrinks by two orders of magnitude.
Figure 3 shows the relative errors of the small angle approximations.
[6] The linear size (D) is related to the angular size (X) and the distance from the observer (d) by the simple formula: where X is measured in arcseconds.
The quantity 206265″ is approximately equal to the number of arcseconds in a circle (1296000″), divided by 2π, or, the number of arcseconds in 1 radian.
The exact formula is and the above approximation follows when tan X is replaced by X.
The second-order cosine approximation is especially useful in calculating the potential energy of a pendulum, which can then be applied with a Lagrangian to find the indirect (energy) equation of motion.
When calculating the period of a simple pendulum, the small-angle approximation for sine is used to allow the resulting differential equation to be solved easily by comparison with the differential equation describing simple harmonic motion.
The sine and tangent small-angle approximations are used in relation to the double-slit experiment or a diffraction grating to develop simplified equations like the following, where y is the distance of a fringe from the center of maximum light intensity, m is the order of the fringe, D is the distance between the slits and projection screen, and d is the distance between the slits: [7]
The small-angle approximation also appears in structural mechanics, especially in stability and bifurcation analyses (mainly of axially-loaded columns ready to undergo buckling).
This leads to significant simplifications, though at a cost in accuracy and insight into the true behavior.
The 1 in 60 rule used in air navigation has its basis in the small-angle approximation, plus the fact that one radian is approximately 60 degrees.
The formulas for addition and subtraction involving a small angle may be used for interpolating between trigonometric table values: Example: sin(0.755)
where the values for sin(0.75) and cos(0.75) are obtained from trigonometric table.
