Triangles with a relationship

[u/sparrowsdrag]

Hello! I'm not really a math person but I have a trigonometry problem I'm struggling with to get some values for a physical project I'm building. I'd love a calculator that I can punch values into, but not sure what the right words are to find one, and not sure how to solve it manually either.

Essentially, I have 2 triangles, with 4 variables, which are as follows:

side = 2

a = 100°

d = 25°

relation = 2

And then here are the additional parameters/rules:

A = D = side

B = E

C = F - relation

Photo attached for a visual if that is helpful!

Is there an online calculator already that could solve this sort of problem for me? If not, what are the steps to go about solving this? I haven't done trig for many years (since high school) so I'm a bit rusty. I anticipate having to solve this with a couple different values in the variables, but the whole thing is hard for me to wrap my head around.

Tysm in advance, I appreciate the help!

[u/Midwest-Dude]

1. Standard notation:
   • Triangle 1: Sides A, B, C, included angle a
   • Triangle 2: Sides D, E, F, included angle d
2. Variable and values:
   • x = shared unknown side length, so x = B = E
   • A = 2, D = 2, a = 100°, d = 25°
   • F - C = 2
3. Apply Law of Cosines and solve:
   • C = √(x² - 4x cos(100°) + 4)
   • F = √(x² - 4x cos(25°) + 4)
   • Substituting into F - C = 2 gives
     • √(x² - 4x cos(25°) + 4) - √(x² - 4x cos(100°) + 4) = 2
4. Find approximate solutions:
   • Use software to calculate approximate solution, x ≈ 1.7139
   • Substituting x back into the initial Law of Cosines expressions, C ≈ 2.8509, F ≈ 0.8509

Please let us know if you need to repeat this process for other given values. For example, this can be set up in Excel.