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

AI Thread Summary
The discussion focuses on determining the angle at which the small angle approximation deviates from the exact trigonometric result by more than 1%. The relevant equations are the approximate formula d = rθ and the exact formula d = 2*r*Sin(θ/2). The user attempts to set up an equation to find θ, using the condition |Exact - Approx.|/Exact = 0.01, leading to (θ/2)*Csc(θ/2) = 1.01. It is noted that this equation can only be solved numerically or graphically, and there is a reminder to convert the angle into degrees. The discussion emphasizes the need for numerical methods to isolate θ accurately.
Airsteve0
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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!
 
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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.
 
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