Textbook Question
Perform the following calculations with the correct number of significant figures.
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1. Intro to Physics Units
Operations with Significant Figures
Problem 9
Textbook Question
For small angles θ, the numerical value of sin θ is approximately the same as the numerical value of tan θ. Find the largest angle for which sine and tangent agree to within two significant figures.
Verified step by step guidance1
Understand the problem: We are tasked with finding the largest angle θ (in radians or degrees) for which the values of sin(θ) and tan(θ) agree to within two significant figures. This means the difference between sin(θ) and tan(θ) should be less than 0.005 (since two significant figures imply a tolerance of ±0.005).
Recall the small-angle approximation: For small angles (measured in radians), sin(θ) ≈ θ and tan(θ) ≈ θ. However, as θ increases, the approximation becomes less accurate, and the difference between sin(θ) and tan(θ) grows. We need to determine the largest θ where the difference is still within the tolerance.
Set up the inequality: The condition for agreement to two significant figures is |sin(θ) - tan(θ)| < 0.005. Using the trigonometric identity tan(θ) = sin(θ)/cos(θ), rewrite the inequality as |sin(θ) - sin(θ)/cos(θ)| < 0.005.
Simplify the inequality: Factor out sin(θ) from the terms inside the absolute value. This gives |sin(θ) * (1 - 1/cos(θ))| < 0.005. Since sin(θ) is positive for small angles, the inequality becomes sin(θ) * |1 - 1/cos(θ)| < 0.005.
Solve numerically or graphically: To find the largest θ satisfying this inequality, you can either solve it numerically using a calculator or graph the functions sin(θ) and tan(θ) to visually determine where their difference is less than 0.005. Ensure the solution is expressed in radians or degrees, depending on the context of the problem.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Small Angle Approximation
The small angle approximation states that for angles measured in radians, the sine and tangent functions can be approximated as equal to the angle itself when the angle is small. This is because as the angle approaches zero, both sin(θ) and tan(θ) approach θ, making them nearly indistinguishable for small values.
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Trigonometric Functions
Sine (sin) and tangent (tan) are fundamental trigonometric functions that relate the angles of a triangle to the ratios of its sides. Specifically, sin(θ) is the ratio of the opposite side to the hypotenuse, while tan(θ) is the ratio of the opposite side to the adjacent side. Understanding these functions is crucial for analyzing their behavior at various angles.
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Significant Figures
Significant figures are the digits in a number that contribute to its precision. When comparing values, such as sin(θ) and tan(θ), determining how closely they agree within a specified number of significant figures helps assess the accuracy of approximations. In this context, finding the largest angle where sin(θ) and tan(θ) are equal to two significant figures is essential for understanding their relationship.
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