That equation can only be solved numerically or graphically.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!
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.
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.
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.
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.
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.