math:trigonometry
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| Both sides previous revisionPrevious revisionNext revision | Previous revision | ||
| math:trigonometry [2025/12/01 21:01] – ron | math:trigonometry [2025/12/01 21:09] (current) – ron | ||
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| given: triangle with an angle **C** with adjacent sides **a** and **b** and opposite side **c**. | 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. | + | * 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** | * 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** | + | * label this new line **x** and the divided parts of **b** as **y** and **z** |
| * **y** + **z** = **b** | * **y** + **z** = **b** | ||
| * from basic Pythagorean we can state the following: | * from basic Pythagorean we can state the following: | ||
| - | * x^2 + z^2 = c^2 | + | * x<sup>2</ |
| - | * x^2 + y^2 = a^2 | + | * x<sup>2</ |
| - | * we can eliminate x^2 easily | + | * we can eliminate x<sup>2</ |
| - | * x^2 = c^2 - z^2 = a^2 - y^2 | + | * x<sup>2</ |
| - | * isolating c^2 | + | * isolating c<sup>2</ |
| - | * c^2 = a^2 - y^2 + z^2 | + | * c<sup>2</ |
| * removing one of the parts of b (since y + z = b then z = b - y) | * removing one of the parts of b (since y + z = b then z = b - y) | ||
| - | * c^2 = a^2 - y^2 + (b - y)^2 | + | * c<sup>2</ |
| - | * expanding (b - y)^2 | + | * expanding (b - y)<sup>2</ |
| - | * c^2 = a^2 - y^2 + b^2 - 2by + y^2 | + | * c<sup>2</ |
| * remove the y^2s | * remove the y^2s | ||
| - | * c^2 = a^2 + b^2 - 2*b*y | + | * c<sup>2</ |
| * from SOH-CAH-TOA we can state that | * from SOH-CAH-TOA we can state that | ||
| - | * cos(C) = y/a so | + | * cos(C) = y/a |
| * y = a * cos(C) | * y = a * cos(C) | ||
| * substitute for y | * substitute for y | ||
| - | * c^2 = a^2 + b^2 - 2 * b * a * cos(C) <-- :) | + | * c<sup>2</ |
math/trigonometry.1764622894.txt.gz · Last modified: 2025/12/01 21:01 by ron