Determine The Percentage Uncertainty In 𝜃, And In Sin 𝜃, When: (a) 𝜃 = 15° ± 0.5° (b) 𝜃 = 75.0° ± 0.5°

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Introduction

In various scientific and engineering applications, it is essential to understand and calculate the uncertainty associated with measured quantities. This is particularly crucial when dealing with trigonometric functions, as small errors in the input angle can lead to significant deviations in the calculated values. In this article, we will explore the concept of percentage uncertainty in trigonometric functions, specifically focusing on the sine function, and apply it to two different scenarios.

What is Percentage Uncertainty?

Percentage uncertainty, also known as relative uncertainty, is a measure of the uncertainty associated with a measured quantity. It is defined as the ratio of the uncertainty to the measured value, expressed as a percentage. Mathematically, it can be represented as:

Percentage Uncertainty = (Uncertainty / Measured Value) × 100

Calculating Percentage Uncertainty in Trigonometric Functions

When dealing with trigonometric functions, the uncertainty in the input angle can propagate to the calculated values. To determine the percentage uncertainty in a trigonometric function, we need to consider the derivative of the function with respect to the input angle.

For the sine function, the derivative is:

d(sin(θ)) / dθ = cos(θ)

Using the chain rule, we can express the uncertainty in the sine function as:

Δ(sin(θ)) = (d(sin(θ)) / dθ) × Δθ = cos(θ) × Δθ

where Δθ is the uncertainty in the input angle.

To calculate the percentage uncertainty in the sine function, we can use the following formula:

Percentage Uncertainty in sin(θ) = (Δ(sin(θ)) / sin(θ)) × 100 = (cos(θ) × Δθ / sin(θ)) × 100

Scenario (a): θ = 15° ± 0.5°

In this scenario, we are given an angle of 15° with an uncertainty of ±0.5°. To calculate the percentage uncertainty in the sine function, we can use the formula above:

Percentage Uncertainty in sin(15°) = (cos(15°) × 0.5° / sin(15°)) × 100

Using a calculator, we can evaluate the trigonometric functions and calculate the percentage uncertainty:

cos(15°) ≈ 0.9659 sin(15°) ≈ 0.2588

Substituting these values into the formula, we get:

Percentage Uncertainty in sin(15°) ≈ (0.9659 × 0.5° / 0.2588) × 100 ≈ 1.86%

Scenario (b): θ = 75.0° ± 0.5°

In this scenario, we are given an angle of 75.0° with an uncertainty of ±0.5°. To calculate the percentage uncertainty in the sine function, we can use the formula above:

Percentage Uncertainty in sin(75.0°) = (cos(75.0°) × 0.5° / sin(75.0°)) × 100

Using a calculator, we can evaluate the trigonometric functions and calculate the percentage uncertainty:

cos(75.0°) ≈ 0.2588 sin(75.0°) ≈ 0.9659

Substituting these values into the formula, we get:

Percentage Uncertainty in sin(75.0°) ≈ (0.2588 × 0.5° / 0.9659) × 100 ≈ 1.34%

Conclusion

In conclusion, we have demonstrated how to calculate the percentage uncertainty in trigonometric functions, specifically the sine function, using the derivative of the function with respect to the input angle. We applied this concept to two different scenarios, where the input angle and its uncertainty were given. The results show that the percentage uncertainty in the sine function can be significant, even for small uncertainties in the input angle. This highlights the importance of considering the uncertainty associated with measured quantities in scientific and engineering applications.

References

  • [1] Introduction to Error Analysis, John R. Taylor, University Science Books, 1997.
  • [2] Trigonometry, Michael Corral, 2003.

Further Reading

  • Propagation of Uncertainty in Trigonometric Functions, Journal of Physics A: Mathematical and Theoretical, Vol. 45, No. 10, 2012.
  • Uncertainty Analysis in Trigonometric Functions, Journal of Engineering Mechanics, Vol. 139, No. 5, 2013.
    Frequently Asked Questions (FAQs) on Percentage Uncertainty in Trigonometric Functions =====================================================================================

Q: What is the purpose of calculating percentage uncertainty in trigonometric functions?

A: Calculating percentage uncertainty in trigonometric functions is essential in scientific and engineering applications to understand the uncertainty associated with measured quantities. This is particularly crucial when dealing with trigonometric functions, as small errors in the input angle can lead to significant deviations in the calculated values.

Q: How do I calculate the percentage uncertainty in a trigonometric function?

A: To calculate the percentage uncertainty in a trigonometric function, you need to consider the derivative of the function with respect to the input angle. The formula for calculating the percentage uncertainty in a trigonometric function is:

Percentage Uncertainty = (Uncertainty / Measured Value) × 100

Q: What is the difference between absolute uncertainty and percentage uncertainty?

A: Absolute uncertainty is the actual value of the uncertainty, while percentage uncertainty is the ratio of the uncertainty to the measured value, expressed as a percentage. Percentage uncertainty is a more useful measure of uncertainty, as it takes into account the size of the measured value.

Q: Can I use the same formula to calculate the percentage uncertainty in all trigonometric functions?

A: No, the formula for calculating the percentage uncertainty in a trigonometric function depends on the specific function. For example, the formula for calculating the percentage uncertainty in the sine function is different from the formula for calculating the percentage uncertainty in the cosine function.

Q: How do I handle the uncertainty in the input angle when calculating the percentage uncertainty in a trigonometric function?

A: When calculating the percentage uncertainty in a trigonometric function, you need to consider the uncertainty in the input angle. This can be done by using the derivative of the function with respect to the input angle, as shown in the formula above.

Q: Can I use a calculator to calculate the percentage uncertainty in a trigonometric function?

A: Yes, you can use a calculator to calculate the percentage uncertainty in a trigonometric function. Most scientific calculators have built-in functions for calculating the derivative of a function, which can be used to calculate the percentage uncertainty.

Q: What are some common applications of percentage uncertainty in trigonometric functions?

A: Percentage uncertainty in trigonometric functions has many applications in scientific and engineering fields, including:

  • Navigation and Surveying: Calculating the uncertainty in the position of a ship or a building using trigonometric functions.
  • Physics and Engineering: Calculating the uncertainty in the values of physical quantities, such as velocity and acceleration, using trigonometric functions.
  • Computer Graphics: Calculating the uncertainty in the values of 3D coordinates using trigonometric functions.

Q: Can I use the same method to calculate the percentage uncertainty in other mathematical functions?

A: Yes, the method for calculating the percentage uncertainty in trigonometric functions can be applied to other mathematical functions, such as exponential and logarithmic functions. However, the formula for calculating the percentage uncertainty will depend on the specific function.

Q: What are some common mistakes to avoid when calculating the percentage uncertainty in trigonometric functions?

A: Some common mistakes to avoid when calculating the percentage uncertainty in trigonometric functions include:

  • Not considering the uncertainty in the input angle: Failing to consider the uncertainty in the input angle can lead to incorrect results.
  • Using the wrong formula: Using the wrong formula for calculating the percentage uncertainty can lead to incorrect results.
  • Not using a calculator or a computer program: Failing to use a calculator or a computer program can lead to errors in the calculation.

Conclusion

In conclusion, calculating the percentage uncertainty in trigonometric functions is an essential task in scientific and engineering applications. By understanding the formula for calculating the percentage uncertainty and avoiding common mistakes, you can ensure accurate results in your calculations.