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Require help understanding small angle approximation

  • Thread starter Kaldanis
  • Start date
  • #1
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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


Unsure


The Attempt at a Solution


Unsure
 

Answers and Replies

  • #2
172
2
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:
[itex]\displaystyle \frac{|sinθ-θ|}{|sinθ|}≤.05[/itex]
 
  • #3
106
0
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:
[itex]\displaystyle \frac{|sinθ-θ|}{|sinθ|}≤.05[/itex]

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

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