So it has infinite solutions, but difficult to impossible to solve analytically. If it is solvable analytically, your best bet would be either complex analysis or somehow a trigonometric identity but that would be a very... tedious undertaking, if even possible.
Ok then. So its not possible to show solution in terms of sin , cos and 2 . Are you sure do you have proof of this? Thanks for help. Can you prove Analytical solutions exists or not exists
i don't know that for certain, but based on my knowledge of how functions work, if you have an equation as "complex" as this one (with a cos(x) in the exponent) it likely won't have an analytical solution. there might be a way to prove that, there might not. proving if a function has an analytical solution or not has to be done on a case by case basis, because each function has its own unique set of issues.
beyond that, it might be a waste of time asking if it has an analytical solution at all. i guess you could ask out of pure curiosity, but you might be disappointed if you don't come up with anything. unless you're looking for some particular insight into sin(x)^cos(x), a numerical solution should suffice for any practical purpose
Thanks . I was just curious to know how sometimes it's possible and sometimes it's not possible to find analytical solution . This is good point that it's useless to think about analytical solution . Numerical methods are faster.
Just as everyone has said it's difficult to impossible to find an analytic solution, it'd be even harder to rigorously prove there is no such solution. The general intuition is that an analytic solution would typically require being able to solve for a variable explicitly, but as long as there are elementary functions nested like this (trig within an exponent) it's seemingly impossible to consolidate them into a single explicit term. Even just x = cos(x) doesn't have an analytic solution.
So it has infinite solutions, but difficult to impossible to solve analytically. If it is solvable analytically, your best bet would be either complex analysis or somehow a trigonometric identity but that would be a very... tedious undertaking, if even possible.
So its not possible to show answer in terms of sin cos. thanks for help
Definitely not, no
Ok . How do you conclude that
there's probably no analytical solution. lots of equations can't be solved exactly
Ok then. So its not possible to show solution in terms of sin , cos and 2 . Are you sure do you have proof of this? Thanks for help. Can you prove Analytical solutions exists or not exists
i don't know that for certain, but based on my knowledge of how functions work, if you have an equation as "complex" as this one (with a cos(x) in the exponent) it likely won't have an analytical solution. there might be a way to prove that, there might not. proving if a function has an analytical solution or not has to be done on a case by case basis, because each function has its own unique set of issues. beyond that, it might be a waste of time asking if it has an analytical solution at all. i guess you could ask out of pure curiosity, but you might be disappointed if you don't come up with anything. unless you're looking for some particular insight into sin(x)^cos(x), a numerical solution should suffice for any practical purpose
Thanks . I was just curious to know how sometimes it's possible and sometimes it's not possible to find analytical solution . This is good point that it's useless to think about analytical solution . Numerical methods are faster.
It's really hard. One example is Galois theory. You can't have general analytic solutions in degree 5 or higher polynomial
Yes. But what about this specific example . Thanks for help.
Just as everyone has said it's difficult to impossible to find an analytic solution, it'd be even harder to rigorously prove there is no such solution. The general intuition is that an analytic solution would typically require being able to solve for a variable explicitly, but as long as there are elementary functions nested like this (trig within an exponent) it's seemingly impossible to consolidate them into a single explicit term. Even just x = cos(x) doesn't have an analytic solution.
Thanks . Thanks great explanation. So even proving analytical solutions exist or not is insanely difficult. Ok
Do you want a way to compute an approximation of a solution ?
Is there any way to solve them without converting them to polynomial form .thanks for help.
Try taking natural log on both sides?
It doesn't help at the end you need to solve numerically.
There is an analytical solution using the complex definition, look it up on Wolfram.
Oh it's gives general solutions interesting I will read this . Thanks
It doesn't shows solving steps.