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--- math:trigonometry [2025/12/01 20:27] – created ron
+++ math:trigonometry [2025/12/01 21:09] (current) – ron
@@ Line 1: / Line 1: @@
-WIP
+# Generalized Pythagorean Equation
+given: triangle with an angle **C** with adjacent sides **a** and **b** and opposite side **c**.
+  * bisect **b** (or **a**) at right angle with line passing thru opposite angle
+    * works easier with obtuse angles such that intersection is internal to triangle - otherwise you end up hanging on the outside of the triangle which feels awkward.
+    * we choose to bisect a side other than **c** so we don't bisect angle **C**
+  * label this new line **x** and the divided parts of **b** as **y** and **z**
+    * **y** + **z** = **b**
+  * from basic Pythagorean we can state the following:
+    * x<sup>2</sup> + z<sup>2</sup> = c<sup>2</sup>
+    * x<sup>2</sup> + y<sup>2</sup> = a<sup>2</sup>
+  * we can eliminate x<sup>2</sup> easily
+    * x<sup>2</sup> = c<sup>2</sup> - z<sup>2</sup> = a<sup>2</sup> - y<sup>2</sup>
+  * isolating c<sup>2</sup>
+    * c<sup>2</sup> = a<sup>2</sup> - y<sup>2</sup> + z<sup>2</sup>
+  * removing one of the parts of b (since y + z = b then z = b - y)
+    * c<sup>2</sup> = a<sup>2</sup> - y<sup>2</sup> + (b - y)<sup>2</sup>
+  * expanding (b - y)<sup>2</sup>
+    * c<sup>2</sup> = a<sup>2</sup> - y<sup>2</sup> + b<sup>2</sup> - 2 * b * y + y<sup>2</sup>
+  * remove the y^2s
+    * c<sup>2</sup> = a<sup>2</sup> + b<sup>2</sup> - 2 * b * y
+  * from SOH-CAH-TOA we can state that
+    * cos(C) = y/a
+    * y = a * cos(C)
+  * substitute for y
+    * c<sup>2</sup> = a<sup>2</sup> + b<sup>2</sup> - 2 * b * a * cos(C) <-- :)