Small-angle approximation

For small <a href="/facts/Angle/gFUTEQYq">angles</a>, the <a href="/facts/Trigonometric_functions/czej7ur0">trigonometric functions</a> 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 <a href="/facts/Radian/Srd38Sxa">radians</a>. Angles measured in <a href="/facts/Degree_(angle)/f48Ns8SX">degrees</a> 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 <a href="/facts/Physics/a7aH2y08">physics</a> and <a href="/facts/Engineering/RFVg8H3I">engineering</a>, including <a href="/facts/Mechanics/A3yHLQA5">mechanics</a>, <a href="/facts/Electromagnetics/oAkmALHT">electromagnetism</a>, <a href="/facts/Optics/p0vq4S1T">optics</a>, <a href="/facts/Cartography/8YdDAAeX">cartography</a>, <a href="/facts/Astronomy/0ArJVM1p">astronomy</a>, and <a href="/facts/Computer_science/lN3pEyPz">computer science</a>. One reason for this is that they can greatly simplify <a href="/facts/Differential_equation/9MGbE3Ca">differential equations</a> 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 <a href="/facts/Maclaurin_series/M0aTuftH">Maclaurin series</a> for each of the trigonometric functions. Depending on the <a href="/facts/Order_of_approximation/FxajXyHk">order of the approximation</a>, 
 
 
 
 
 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}}
 
.

Small-angle approximation open-in-new

Small-angle approximation