Require help understanding small angle approximation

In summary, the conversation discusses the use of approximations in physics and provides an example of using sinθ and tanθ to simplify problems. The question then asks for the range of values for θ where the errors in approximating sinθ and tanθ by θ are less than 5%. The equation |sinθ-θ|/|sinθ| is used to calculate the error.
  • #1
Kaldanis
106
0

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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  • #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
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:
[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.
 

1. What is the small angle approximation?

The small angle approximation is a mathematical technique used to approximate the value of trigonometric functions for very small angles. It assumes that for angles close to zero, the sine and tangent of the angle can be approximated by the angle itself, while the cosine can be approximated by 1.

2. Why is the small angle approximation useful?

The small angle approximation is useful because it simplifies calculations involving trigonometric functions for small angles. It allows for easier and quicker calculations, especially in situations where highly accurate results are not necessary.

3. How accurate is the small angle approximation?

The accuracy of the small angle approximation depends on the size of the angle being approximated. For angles close to zero, the approximation is very accurate, but as the angle increases, the accuracy decreases. It is important to note that the small angle approximation is an estimate and not an exact value.

4. Can the small angle approximation be used for any angle?

No, the small angle approximation is only valid for very small angles. As the angle increases, the error in the approximation also increases. It is generally recommended to use the small angle approximation for angles less than 15 degrees.

5. How is the small angle approximation derived?

The small angle approximation is derived using the Taylor series expansion of trigonometric functions. By keeping only the first two terms of the series, the trigonometric functions can be approximated by the angle itself. This is only possible for small angles, as the higher order terms become more significant as the angle increases.

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