The accuracy of the approximations can be seen below in Figure 1 and Figure 2. As the measure of the angle approaches zero, the difference between the approximation and the original function also approaches 0.
Geometric
The red section on the right,, is the difference between the lengths of the hypotenuse,, and the adjacent side,. As is shown, and are almost the same length, meaning is close to 1 and helps trim the red away. The opposite leg,, is approximately equal tothe length of the blue arc,. Gathering facts from geometry,, from trigonometry, and, and from the picture, and leads to: Simplifying leaves,
Calculus
Using the squeeze theorem, we can prove that which is a formal restatement of the approximation for small values of θ. A more careful application of the squeeze theorem proves that from which we conclude that for small values of θ. Finally, using L'Hôpital's rule tells us that which rearranges algebraically to give for small values of θ.
Algebraic
The Maclaurin expansion of the relevant trigonometric function is where is the angle in radians. In clearer terms, It is readily seen that the second most significant term falls off as the cube of the first term; thus, even for a not-so-small argument such as 0.01, the value of the second most significant term is on the order of, or the first term. One can thus safely approximate: By extension, since the cosine of a small angle is very nearly 1, and the tangent is given by the sine divided by the cosine,
Error of the approximations
Figure 3 shows the relative errors of the small angle approximations. The angles at which the relative error exceeds 1% are as follows:
In astronomy, the angular size or angle subtended by the image of a distant object is often only a few arcseconds, so it is well suited to the small angle approximation. The linear size is related to the angular size and the distance from the observer by the simple formula: where is measured in arcseconds. The number is approximately equal to the number of arcseconds in a circle, divided by. The exact formula is and the above approximation follows when is replaced by.
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 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 small-angle approximation also appears in structural mechanics, especially in stability and bifurcation analyses. This leads to significant simplifications, though at a cost in accuracy and insight into the true behavior.
Piloting
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.
Interpolation
The formulas for [|addition and subtraction involving a small angle] may be used for interpolating between trigonometric table values: Example: sin
