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Determining the extent of a measurement and expressing it with a specific degree of precision, in this instance, hundredths of a unit, is a common requirement in various fields. For example, when machining a component, one might need to specify a dimension of 25.43 millimeters rather than simply 25 millimeters or 25.4 millimeters. This provides a higher level of accuracy, suitable for applications requiring tight tolerances.

The ability to establish a dimension with this level of specificity is crucial for ensuring compatibility, functionality, and safety in numerous applications. Historically, achieving this precision relied on meticulous manual measurements and calculations. Now, instruments and software enable efficient and accurate determination. This has resulted in improvements in manufacturing, engineering, construction, and scientific research by providing greater control and consistency.

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Determining the cosine of an angle and expressing the result with a precision of two decimal places involves a trigonometric calculation followed by a rounding operation. The cosine function, a fundamental concept in trigonometry, relates an angle in a right triangle to the ratio of the adjacent side to the hypotenuse. For example, the cosine of 60 degrees is 0.5. Expressing this value to two decimal places would be 0.50.

The significance of obtaining trigonometric values with limited decimal places lies in its practical application in various fields. In engineering, architecture, and physics, measurements and calculations often require a balance between accuracy and simplicity. Rounding to two decimal places offers a reasonable level of precision for many real-world problems, while avoiding unnecessary complexity. Historically, this level of approximation has been adequate for tasks where minute variations are negligible, allowing for easier manual computations before the widespread availability of calculators and computers.

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Determining the measure of side ‘b’ within a geometric figure, specifically to a precision of hundredths, involves employing mathematical principles such as the Pythagorean theorem, trigonometric ratios (sine, cosine, tangent), or the law of sines/cosines, depending on the information available about the figure. For instance, in a right triangle where side ‘a’ is 3 units, the hypotenuse ‘c’ is 5 units, side ‘b’ can be calculated using the Pythagorean theorem (a + b = c), resulting in b = (c – a) = (25 – 9) = 16 = 4.00 units.

Precision in measurement is crucial across various scientific, engineering, and construction disciplines. Accurate dimensioning prevents errors in design, fabrication, and assembly processes. Specifying the result to two decimal places offers a balance between accuracy and practicality, ensuring sufficient detail for most applications while avoiding excessive computational complexity or the need for extremely precise measuring instruments. Historically, the need for precise dimensional calculations has driven advancements in mathematics and measurement technologies.

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