When does the small angle approximation deviate by more than 1%?

In summary, the homework problem asks to find the angle above which the small angle approximation differs from the exact result by more than 1 percent. The equation for this is given as (θ/2)*Csc(θ/2) = 1.01, and it can only be solved numerically or graphically. It is also important to remember to convert the angle to degree measure.
  • #1
Airsteve0
83
0

Homework Statement


Find, by comparison with exact trigonometry, the angle,  (provide a numerical value
in degrees), above which the small angle approximation departs from the exact result by more than 1 percent.


Homework Equations



Approx.: d = s = rθ
Exact: d = 2*r*Sin(θ/2)

The Attempt at a Solution



.01 = |Exact - Approx.|/Exact

(θ/2)*Csc(θ/2) = 1.01


At this point I am unsure of how to isolate for θ. Any tips are greatly appreciated, thanks!
 
Physics news on Phys.org
  • #2
Airsteve0 said:

Homework Statement


Find, by comparison with exact trigonometry, the angle,  (provide a numerical value
in degrees), above which the small angle approximation departs from the exact result by more than 1 percent.


Homework Equations



Approx.: d = s = rθ
Exact: d = 2*r*Sin(θ/2)

The Attempt at a Solution



.01 = |Exact - Approx.|/Exact

(θ/2)*Csc(θ/2) = 1.01

At this point I am unsure of how to isolate for θ. Any tips are greatly appreciated, thanks!
That equation can only be solved numerically or graphically.

Don't forget to change the angle to degree measure.
 

FAQ: When does the small angle approximation deviate by more than 1%?

What is the Small Angle Approximation?

The Small Angle Approximation is a mathematical technique used to simplify trigonometric functions when the angle is small, typically less than 10 degrees. It is based on the idea that for small angles, the sine of the angle is approximately equal to the angle itself in radians.

Why is the Small Angle Approximation useful?

The Small Angle Approximation is useful because it allows us to simplify complex trigonometric expressions and make them easier to work with. It also allows us to make accurate approximations without having to use a calculator or perform lengthy calculations.

What are the limitations of the Small Angle Approximation?

The Small Angle Approximation is only useful for small angles, typically less than 10 degrees. Beyond this range, the error in the approximation becomes too large and can lead to inaccurate results. Additionally, the Small Angle Approximation is only suitable for certain types of trigonometric functions, such as sine and cosine.

How is the Small Angle Approximation used in science?

The Small Angle Approximation is used in various fields of science, including physics, astronomy, and engineering. It is often used to simplify equations and make calculations easier, especially when dealing with small angles in geometric or kinematic problems. It is also used in optics, where small angles are common in the study of light and its behavior.

Can the Small Angle Approximation be applied to any angle?

No, the Small Angle Approximation is only valid for small angles. For larger angles, a more accurate mathematical approach must be used. Additionally, the Small Angle Approximation is only applicable to certain types of trigonometric functions and cannot be used for all types of mathematical expressions.

Similar threads

Small Angle Approximation in Single Slit Interference
Replies
3
Views
7K
Calculating the capacitance of a system of two plates tilted at a small angle
Replies
2
Views
1K
Require help understanding small angle approximation
Replies
2
Views
9K
Finding the angle of phase difference - two slit model
Replies
7
Views
4K
Deriving the Time Period of a Pendulum with Small Angle Approximation
Replies
2
Views
1K
What is the value of θ when the error in sinθ ≈ θ is approximately 5%?
Replies
5
Views
6K
Angle of Incidence, Propagation Constant and Wavelength
Replies
1
Views
2K
Single-slit Diffraction Problem
Replies
4
Views
2K
A How Does U(1) Double-Cover SO(2) for a Specified Angle?
Replies
2
Views
1K
Far Field Approximation After a Prism
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top