For small angles, the trigonometric functions sine, cosine, and tangent can be calculated with reasonable accuracy by the following simple approximations:
sin θ ≈ tan θ ≈ θ , cos θ ≈ 1 − 1 2 θ 2 ≈ 1 , {\displaystyle {\begin{aligned}\sin \theta &\approx \tan \theta \approx \theta ,\\[5mu]\cos \theta &\approx 1-{\tfrac {1}{2}}\theta ^{2}\approx 1,\end{aligned}}}
provided the angle is measured in radians. Angles measured in degrees must first be converted to radians by multiplying them by π / 180 {\displaystyle \pi /180} . 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, cos θ {\displaystyle \textstyle \cos \theta } is approximated as either 1 {\displaystyle 1} or as 1 − 1 2 θ 2 {\textstyle 1-{\frac {1}{2}}\theta ^{2}} .
Justifications
Geometric
For a small angle, H and A are almost the same length, and therefore cos θ is nearly 1. The segment d (in red to the right) is the difference between the lengths of the hypotenuse, H, and the adjacent side, A, and has length H − H 2 − O 2 {\displaystyle \textstyle H-{\sqrt {H^{2}-O^{2}}}} , which for small angles is approximately equal to O 2 / 2 H ≈ 1 2 θ 2 H {\displaystyle \textstyle O^{2}\!/2H\approx {\tfrac {1}{2}}\theta ^{2}H} . As a second-order approximation,
cos θ ≈ 1 − θ 2 2 . {\displaystyle \cos {\theta }\approx 1-{\frac {\theta ^{2}}{2}}.}
The opposite leg, O, is approximately equal to the length of the blue arc, s. The arc s has length θA, and by definition sin θ = O/H and tan θ = O/A, and for a small angle, O ≈ s and H ≈ A, which leads to:
sin θ = O H ≈ O A = tan θ = O A ≈ s A = A θ A = θ . {\displaystyle \sin \theta ={\frac {O}{H}}\approx {\frac {O}{A}}=\tan \theta ={\frac {O}{A}}\approx {\frac {s}{A}}={\frac {A\theta }{A}}=\theta .}
Or, more concisely,
sin θ ≈ tan θ ≈ θ . {\displaystyle \sin \theta \approx \tan \theta \approx \theta .}
Calculus Using the squeeze theorem, one can prove that
… excerpt ends here. Continue reading the full article.
