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Concept
Small-angle approximation
Summary
The small-angle approximations can be used to approximate the values of the main trigonometric functions, provided that the angle in question is small and is measured in radians:
These approximations have a wide range of uses in branches of physics and engineering, including mechanics, electromagnetism, optics, cartography, astronomy, and computer science. 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. Depending on the order of the approximation, is approximated as either or as .
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.
File:Small_angle_compair_odd.svg|'''Figure 1.''' A comparison of the basic [[odd function|odd]] trigonometric functions to {{mvar|θ}}. It is seen that as the angle approaches 0 the approximations become better.
File:Small_angle_compare_even.svg|'''Figure 2.''' A comparison of {{math|cos ''θ''}} to {{math|1 − {{sfrac|''θ''2|2}}}}. It is seen that as the angle approaches 0 the approximation becomes better.
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:
Simplifying leaves,
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.
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