# Trigonometric equation solving / 4th degree polynomial

Okay, so I was solving a free body diagram problem, no need to send it, but I found a very huge problem in doing so, in any way I tried solving , I either got to an equation $\sin(x-0.11)=4\sin(2x)$ or a 4th degree polynomial that my calculator wont possibly solve. Any idea on any of the two and a general rule for the sine at least?

• It doesn't look solvable to me. – Simply Beautiful Art Feb 24 '16 at 21:00

Using angle subtraction formula for the left and double angle formula for the right:

$$\sin(x)\cos(0.11)-\sin(0.11)\cos(x)=8\sin(x)\cos(x)$$

• but this also wont help in giving x – Mike Harb Feb 24 '16 at 19:30

Okay, so I was solving a free body diagram problem

Aha, say no more! A physics problem! :-$)$ This means that you'll probably settle for a high precision numerical approximation, since physicists, engineers, economists, and others who study applied mathematics1 aren't exactly famous for their burning interest in closed form expressions, as those noble minds that pursue the higher spheres of pure mathematics2 $($which under no circumstance whatsoever is to imply that other forms of mathematics are in any way, shape or manner impure $\ldots$ or so my PR people are telling me to write $\ldots$ since we're all so PC, tolerant, open-minded, and all-inclusive around here $\ldots)$

1 Politically correct term for “impure mathematics”. Also politically incorrect, since it automatically implies that other branches are to some extent useless. See following footnote.

2 Politically correct term for “useless mathematics”. Also politically incorrect, since it automatically implies that other branches are to some extent impure. See previous footnote.

$($It's getting harder and harder with each passing day to keep track of all those hurt feelings and offended sensibilities, not to mention possible grounds for future lawsuits $\ldots)$

Without further ado, the three main solutions for $|x|<\dfrac\pi2~($since this is the principal interval of the arcsine function$)$ are

$\qquad\qquad\qquad\qquad\quad$

Don't get me wrong, I've tried evaluating the general expression for $~\cos(a)~t-\sin(a)\sqrt{1-t^2}$ $=~8t\sqrt{1-t^2},~$ as well as using the fact that $\sin a\simeq a$ and $\cos a\simeq1$ or $1-\dfrac{a^2}2$ for small values of the argument, but they only yielded hideous expressions, which remained ugly even after applying further simplifications. Replacing a with $\dfrac19$ or $\dfrac{11}{100}$ did not work any wonders either; all it did was return the quartic equivalent of the casus irreducibilis. So, to finally answer your question about finding “a general rule for the sine”, my reply would be to try a numerical approach $\ldots$ and please trust when I tell you that I am not the kind of person to dispense such advice lightly $\ldots$ The first two solutions are very close to $~\pm\dfrac{13}9,~$ and the third can be approximated by $0.01567$ : and that's about as helpful as I can get, I'm afraid $\ldots$

• You make my day with your final recommendation !! Cheers. – Claude Leibovici Feb 25 '16 at 12:43
• So physically I couldnt get anywhere farther than this equation as a mathematical approach ... Ill just do a trail and error technique , thank you very much – Mike Harb Feb 25 '16 at 16:40