u/FanBitter3365

Just for reference, the first image is the triangle, the second image is my work (sorry about how chaotic it is lol) and the third is my answer check.

My teacher gave this question as a kind of "think-on-it" on the final back in my senior year of high school, but I wasn't able to complete it in class. I made note of it on my phone and it got buried there for a while.

Fast forward to today, I'm in my College Algebra course, and looking through my notes I made for that particular course's final, I found that note from my Precalc class again. After the semester ended, math-fresh and equipped with my prior knowledge, I decided to take a crack at it after two years.

This took a lot more time than I thought it would, and given the mess that is my work, I'm almost positive I made a mistake. I haven't done Precalc-related trig stuff in the time of those two years, so I am a little rusty. If somebody could help me realize what I did wrong, I would be grateful.

*Note: this will probably anger some math educators, so I apologize in advance. In my defense, at this point, I was thoroughly mentally exhausted, and I was not about to derive what x is myself when I got to

x = arcsin (√[2 + √3] / 2)

and

x = arcsin (√[2 - √3] / 2)

respectively without using a calculator. That's where I got this from:

x = arcsin (√[2 + √3] / 2)

x = 75°

and

x = arcsin (√[2 - √3] / 2)

x = 15°

Edit: if anyone needs more information, please feel free to ask :)

This has been on my mind for at least a week or two now (I apologize for the crude photo). I tried to compute if it is possible for positive and real x values to make the triangle work, but it ended up being a lot more work (the highest math I have taken is precalculus so my knowledge is limited) and I am ready to call it quits. Any ideas?

I wanted to see if a triangle with legs sin(2x) and sin (3x) and a hypotenuse of sin(4x) was possible. I believe I messed up at some point in my computation (see third image lol my work is chaos), particularly dropping the negative solutions towards the end. Any ideas about what I did wrong? Someone please help.