Insights Physical Applications of the “Tan Rule”

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The discussion highlights the "tan rule," which is often overlooked in trigonometry education despite its utility in solving triangles. It is particularly effective when two sides and the included angle are known, allowing for a more straightforward calculation of the remaining angles. Unlike the sine and cosine rules, which require additional steps to find the third side before proceeding, the tan rule simplifies the process. The article aims to demonstrate the practical applications of this rule in various scenarios. Overall, the tan rule can enhance problem-solving efficiency in trigonometry.
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Introduction
Every secondary school student who has encountered trigonometry in his/her Math syllabus will most likely have come across the sine, cosine, and area rules which are typically used to solve triangles in which certain information is supplied and the remainder are to be calculated. Somewhat surprisingly (because it is relatively simple to derive), the “tan rule” is generally not included as part of this particular set of trig tools. Yet, as we hope to demonstrate in this article, this rule can be extremely useful in certain circumstances. Specifically in the instance where two sides and an included angle of a triangle are given.  In this situation, it is impossible to immediately apply the sine rule to determine the remaining angles since neither of the given sides is opposite the given angle. The cosine rule must first be applied to determine the third side and thereafter the sine rule for either (or both) of the remaining angles. However, the tan rule enables us to...

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Thank you for a very informative article
 
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Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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