Require help understanding small angle approximation

AI Thread Summary
The discussion centers on understanding the small angle approximation in physics, specifically for sinθ and tanθ. The user seeks clarification on calculating the angles where the error in these approximations is approximately 5%. They initially struggled with the concept but found the correct approach after realizing they were using the error equation incorrectly. The key equation for determining the error is |sinθ - θ| / |sinθ|, which helps quantify the difference between the actual value and the approximation. With this understanding, the user successfully solved the initial questions.
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Homework Statement


"In order to simplify problems in physics, we often use various approximations. For example, when we investigate diffraction and interference patterns at small angles θ, we frequently approximate sinθ and tanθ by θ (in radians). Here you will calculate over what range these are reasonable approximations.

For θ= 43° this approximation has an error of almost exactly 10%:

θ = 43.0° = 0.75 radians

sinθ=0.682

|sinθ-θ| / |sinθ| ≈ 10%"

1) For what value of θ (to the nearest degree) is the error in sinθ ≈ θ approximately 5%?
2) For what value of θ (to the nearest degree) is the error in tanθ ≈ θ approximately 5%?

I was recently given this question and very little explanation of the concept. I've struggled with this for a week and read absolutely everything I can find and I'm still not any closer to understanding it. Can anyone please point me in the right direction or explain how to do question 1) and 2)? There are many more questions, but if I can get 1) and 2) down then I should be able to answer the rest by myself. I appreciate any help.

Homework Equations


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The Attempt at a Solution


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The equation at the end explains that |sinθ-θ|/|sinθ| gives the error in approximating sinθ by θ. In other words, it gives the percentage which theta differs from sinθ.

The first question asks: For what values of theta does theta differ from sinθ by less than 5%? in other words:
\displaystyle \frac{|sinθ-θ|}{|sinθ|}≤.05
 
Nessdude14 said:
The equation at the end explains that |sinθ-θ|/|sinθ| gives the error in approximating sinθ by θ. In other words, it gives the percentage which theta differs from sinθ.

The first question asks: For what values of theta does theta differ from sinθ by less than 5%? in other words:
\displaystyle \frac{|sinθ-θ|}{|sinθ|}≤.05


Thank you, I had been using the equation incorrectly without realising. I have the correct answers now.
 
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