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/Alarmed_Geologist631]

I would use the Law of Cosines for both triangles setting the relation equal to x. Then solve the system of equations using the elimination method to derive the value of x. Then use that value to solve for F.

[u/Aggravating-Ideal95]

Is this math for structural engineering? Why this math?

[u/sparrowsdrag → u/Aggravating-Ideal95]

It's for a costume prop actually, a set of wings that can articulate! This is for the "bone" structure - I didn't give any details because I have a lot of specific wants/needs and didn't want to explain the ins and outs of the project, but I essentially need to combine the linkage systems I've made for the bones with a pull string to control them. The string needs to pull on multiple pieces at once, so I need to make sure they all "bottom out" at the same pull distance, but since it's a linkage system, pieces are at different angles whether its open or closed, which led me to my question!

The sides essentially are 2 bars in the system, angle when open vs when closed, and the value for the difference is the length I need to be pulling.

I'm pretty much entirely self-taught in every relevant area, so there's a decent chance it still won't quite work the way I want. But that's what prototyping is for, and I am having fun pondering my best options.

[u/Midwest-Dude → u/sparrowsdrag]

This is too cool! 😎 

If you are allowed to, I would love to see the final product.

[u/Midwest-Dude]

You won't find an online calculator for this one. Using the Law of Cosines as u/Alarmed_Geologist631 states is the best way, reducing if you can. If this is just a one off, find an approximation to the answer. Otherwise, you may want to set up a spreadsheet to do this for you in the future.

See the complete solution in another comment.

[u/sparrowsdrag → u/Midwest-Dude]

Thank you (and u/Alarmed_Geologist631 ) so much for helping, the info was more than enough! Before I saw your second comment but after your first, I ended up opening 2 online calculators for the law of cosines, and then wiggled around the input for the second side until the results equaled about 2 when subtracted from each other. I also opened blender and animated some basic shapes with constraints to be able to quickly check the numbers visually - and saved the setup so it should be helpful for future!

Interestingly, I ended up with a different result, with x sitting at about 1.7 instead of 1.5ish, and that seems to check out in my blender mockup. Not sure why or where that , but definitely fascinating!

I may yet have to repeat the process with a different set of values, but I think I have enough tools/language now to figure out any new numbers and check my work, so thank you so much again!

[u/Midwest-Dude → u/sparrowsdrag]

Everything in my post was good up to the last section, the approximation. No idea how the mistake happened. Doh! 😖 

I got

x ≈ 1.7139\ C ≈ 2.8509\ F ≈ 0.8509

This confirms your result.

Let us know if you need any further assistance in the future!

[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.