What constitutes a small angle?

Question

What constitutes a small angle?  At what angle does the approximation sin(angle) ≅angle fail?  Determine angle such that sin angle and angle differ by 3%.  Hint, must do this in radians......

Answer 1

Megan,

 

Just do a simple table comparing the angle Θ in radians to sin(Θ) from your calculator. To convert from degrees to radians divide by 57.2958.  Here's the table:

 

Θ (degrees)    Θ (radians)          sin(Θ)          [Θ /sin(Θ)]  

_________    __________     _________   ____________

 

10                  0.17453            0.17365          1.0051          (so Θ is 1/2 of 1% high)

15                  0.26180            0.25882          1.0115          (so Θ is 1.15% high)

20                  0.34906            0.34202          1.0206          (so Θ is 2.06% high)

24                  0.41888            0.40674          1.0299          (so Θ is 3.00% high)

25                  0.43633            0.42262          1.0324          (so Θ is 3.24% high)

30                  0.52360            0.50000          1.0472          (so Θ is 4.72% high)

 

Your 3% solution is shown in the table as a 24 degree angle. It comes from using interpolation to solve the relationship 

[Θ /sin(Θ)] = 1.03.

 

Thus, how small an angle should be to replace sin(Θ) with Θ (in radians, not degrees) depends on how much error you're willing to tolerate.  The error for 15 degrees is barely more than 1%, so that's a good figure.

 

BTW, such an approximation is used in Physics to solve pendulum motion where the pull of gravity is approximated by the restoring force -KΘ.