Small-angle approximation

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.

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  Figure 1. A comparison of the basic odd trigonometric functions to θ. It is seen that as the angle approaches 0 the approximations become better.
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  Figure 2. A comparison of cos θ to 1 − ⁠θ²/2⁠. 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. $${\displaystyle \cos {\theta }\approx 1-{\frac {\theta ^{2}}{2}}}$$ 

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: $${\displaystyle \sin \theta ={\frac {O}{H}}\approx {\frac {O}{A}}=\tan \theta ={\frac {O}{A}}\approx {\frac {s}{A}}={\frac {A\theta }{A}}=\theta .}$$ 

Simplifying leaves, $${\displaystyle \sin \theta \approx \tan \theta \approx \theta .}$$ 

Using the squeeze theorem,^[4] we can prove that $${\displaystyle \lim _{\theta \to 0}{\frac {\sin(\theta )}{\theta }}=1,}$$  which is a formal restatement of the approximation ${\displaystyle \sin(\theta )\approx \theta }$  for small values of θ.

A more careful application of the squeeze theorem proves that $${\displaystyle \lim _{\theta \to 0}{\frac {\tan(\theta )}{\theta }}=1,}$$  from which we conclude that ${\displaystyle \tan(\theta )\approx \theta }$  for small values of θ.

Finally, L'Hôpital's rule tells us that $${\displaystyle \lim _{\theta \to 0}{\frac {\cos(\theta )-1}{\theta ^{2}}}=\lim _{\theta \to 0}{\frac {-\sin(\theta )}{2\theta }}=-{\frac {1}{2}},}$$  which rearranges to ${\textstyle \cos(\theta )\approx 1-{\frac {\theta ^{2}}{2}}}$  for small values of θ. Alternatively, we can use the double angle formula ${\displaystyle \cos 2A\equiv 1-2\sin ^{2}A}$ . By letting ${\displaystyle \theta =2A}$ , we get that ${\textstyle \cos \theta =1-2\sin ^{2}{\frac {\theta }{2}}\approx 1-{\frac {\theta ^{2}}{2}}}$ .

 

The Maclaurin expansion (the Taylor expansion about 0) of the relevant trigonometric function is^[5] $${\displaystyle {\begin{aligned}\sin \theta &=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}\theta ^{2n+1}\\&=\theta -{\frac {\theta ^{3}}{3!}}+{\frac {\theta ^{5}}{5!}}-{\frac {\theta ^{7}}{7!}}+\cdots \end{aligned}}}$$  where θ is the angle in radians. In clearer terms, $${\displaystyle \sin \theta =\theta -{\frac {\theta ^{3}}{6}}+{\frac {\theta ^{5}}{120}}-{\frac {\theta ^{7}}{5040}}+\cdots }$$ 

It is readily seen that the second most significant (third-order) 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 0.000001, or ⁠1/10000⁠ the first term. One can thus safely approximate: $${\displaystyle \sin \theta \approx \theta }$$ 

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, $${\displaystyle \tan \theta \approx \sin \theta \approx \theta ,}$$ 

One may also use dual numbers, defined as numbers in the form ${\displaystyle a+b\varepsilon }$ , with ${\displaystyle a,b\in \mathbb {R} }$  and ${\displaystyle \varepsilon }$  satisfying by definition ${\displaystyle \varepsilon ^{2}=0}$  and ${\displaystyle \varepsilon \neq 0}$ . By using the MacLaurin series of cosine and sine, one can show that ${\displaystyle \cos(\theta \varepsilon )=1}$  and ${\displaystyle \sin(\theta \varepsilon )=\theta \varepsilon }$ . Furthermore, it is not hard to prove that the Pythagorean identity holds:$${\displaystyle \sin ^{2}(\theta \varepsilon )+\cos ^{2}(\theta \varepsilon )=(\theta \varepsilon )^{2}+1^{2}=\theta ^{2}\varepsilon ^{2}+1=\theta ^{2}\cdot 0+1=1}$$ 

 

Figure 3 shows the relative errors of the small angle approximations. The angles at which the relative error exceeds 1% are as follows:

• cos θ ≈ 1 at about 0.1408 radians (8.07°)
• tan θ ≈ θ at about 0.1730 radians (9.91°)
• sin θ ≈ θ at about 0.2441 radians (13.99°)
• cos θ ≈ 1 − ⁠θ²/2⁠ at about 0.6620 radians (37.93°)

The angle addition and subtraction theorems reduce to the following when one of the angles is small (β ≈ 0):

cos(α + β)	≈ cos(α) − β sin(α),
cos(α − β)	≈ cos(α) + β sin(α),
sin(α + β)	≈ sin(α) + β cos(α),
sin(α − β)	≈ sin(α) − β cos(α).

In astronomy, the angular size or angle subtended by the image of a distant object is often only a few arcseconds (denoted by the symbol ″), so it is well suited to the small angle approximation.^[6] The linear size (D) is related to the angular size (X) and the distance from the observer (d) by the simple formula:

${\displaystyle D=X{\frac {d}{206\,265{''}}}}$ 

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

${\displaystyle D=d\tan \left(X{\frac {2\pi }{1\,296\,000{''}}}\right)}$ 

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.

In optics, the small-angle approximations form the basis of the paraxial approximation.

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]$${\displaystyle y\approx {\frac {m\lambda D}{d}}}$$ 

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) $${\displaystyle {\begin{aligned}\sin(0.755)&=\sin(0.75+0.005)\\&\approx \sin(0.75)+(0.005)\cos(0.75)\\&\approx (0.6816)+(0.005)(0.7317)\\&\approx 0.6853.\end{aligned}}}$$  where the values for sin(0.75) and cos(0.75) are obtained from trigonometric table. The result is accurate to the four digits given.

• Skinny triangle
• Small oscillations of a pendulum
• Versine and haversine
• Exsecant and excosecant