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 'Erroneously finding discrepancy in transpose rule'

Thread 'Erroneously finding discrepancy in transpose rule'

Obviously, there is something elementary I am missing here. To form the transpose of a matrix, one exchanges rows and columns, so the transpose of a scalar, considered as (or isomorphic to) a one-entry matrix, should stay the same, including if the scalar is a complex number. On the other hand, in the isomorphism between the complex plane and the real plane, a complex number a+bi corresponds to a matrix in the real plane; taking the transpose we get which then corresponds to a-bi...
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