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Trigonometry question?

  • 26-07-2025 03:50PM
    #1
    rock22
    Registered Users, Registered Users 2 Posts: 1,834 ✭✭✭


    Not sure if it is allowed to ask matehmatic questions. If not then mods please delete.

    It has been over fifty years since I used trigonometry, I remember sin, cos , tan etc and Pythagoras theorem which usually allows me to work these questions out but i am stumped here. This layout is from design of two bevel gears but I am not too sure how to solve.

    TrigonometryQuestion.jpg

    Essentially ABC and ACD are two right angle triangles (angles at B and D are 90 degrees). which share the same hypothenuse, h. BC is 37 mm long, CD is 75 mm long. DC is at 10 degrees from vertical so the total angle BCD is 80 degrees. That is all the information I have.

    I need to calculate both the angles c1 and c2. (i.e. angle BCA and angle ACD ) and also the length X and Y ( i.e. AB and AD).

    I know all the sin cos tan relationships but i cannot find a way to use the information i have to make the calculations . Any help or pointer in the right direction would be appreciated. Thanks

    PS. I know I can probably take measurements from the drawing but I would prefer to be able to calculate using trigonometry.



Comments

  • #2
    Michael Collins
    Moderators, Science, Health & Environment Moderators Posts: 1,855 Mod ✭✭✭✭Michael Collins


    Since the hypotenuse is common to both, you could say that:

    h = 37/cos(c1) = 75/cos(c2)

    We also know that c1 + c2 = 80. This now gives

    h = 37/cos(c1) = 75/cos(80° - c1)

    which is one equation in one unknown. Unfortunately it doesn't have a "nice" simple closed-form solution, but we can solve numerically:

    https://www.wolframalpha.com/input?i=37%2Fcos%28x%29+%3D+75%2F%28cos%2880deg-x%29%29

    This gives the acute angle c1 as 1.0824 rad or 62.02°. Can also work out c2 and h quite easily now.



  • #3
    Jacovs
    Registered Users, Registered Users 2 Posts: 2,137 ✭✭✭Jacovs


    And to confirm the maths above…

    Screenshot 2025-08-07 122811.png


  • #4
    Michael Collins
    Moderators, Science, Health & Environment Moderators Posts: 1,855 Mod ✭✭✭✭Michael Collins


    Very nice, Jacovs. What software is that from? Does it just draw it or did it also work out the angles for you?



  • #5
    Jacovs
    Registered Users, Registered Users 2 Posts: 2,137 ✭✭✭Jacovs


    Thats just regular AutoCad. You input the known information to draw the lines. Then you just use the dimension function to indicate all the various angles and lengths for you.



  • #6
    Michael Collins
    Moderators, Science, Health & Environment Moderators Posts: 1,855 Mod ✭✭✭✭Michael Collins


    Excellent, thanks.



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  • #7
    rock22
    Registered Users, Registered Users 2 Posts: 1,834 ✭✭✭rock22


    Thanks for the solutions

    https://www.boards.ie/discussion/comment/123703271#Comment_123703271

    Thanks Michael

    I had worked out as far as your second equation

    h = 37/cos(c1) = 75/cos(80° - c1)

    But your last still leaves me stumped. I am unsure what is Wolfram|Alpha

    Thanks Jacovs for the diagram. I eventually created a similar drawing in FreeCAD. My purpose in making the calculations was to calculate meshing bevel gears for the top of a traditional windmill and I really wanted a mathematical solution in order to easily remake calculations if the mesh angle changes

    I eventually found this equation

    image.png

    where the angle δ1 is the complement 90-c2 and δ2 is 90 - c1. Ʃ = 100 ° (δ1 + δ2) and Z1 = 75, Z2 = 37

    This came from an engineer workbook. Not too sure if it is mathematically correct or an approximation but calculating

    δ1 and δ2 and then subtraction from 90 gives me the same result as Jacovs diagram ( and my freeCAD) at

    c1 = 90 - 27.98430189° = 62.01569811 °

    C2 = 90 - 72.01569808 ° = 17.98430192 °

    Thanks Michael And Jacovs for your help.



  • #8
    rock22
    Registered Users, Registered Users 2 Posts: 1,834 ✭✭✭rock22


    And sorry for being so slow to respond to both your solutions



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