# Thread: [SOLVED] equation with trigonometric and linear variable

1. ## [SOLVED] equation with trigonometric and linear variable

I wasn't sure where to post this as i'm not sure if it's blindingly simple or more difficult, I'm an out of practice engineering student so either is just as likely
anyway, just trying to solve for theta
$\displaystyle 3 \theta - \sin \theta = 2 \pi$

I could easily get an answer using a calculator, but i want to do it analytically.

2. Originally Posted by dinNA89
I wasn't sure where to post this as i'm not sure if it's blindingly simple or more difficult, I'm an out of practice engineering student so either is just as likely
anyway, just trying to solve for theta
$\displaystyle 3 \theta - \sin \theta = 2 \pi$

I could easily get an answer using a calculator, but i want to do it analytically.
Not sure but this might help you
$\displaystyle Let f(\theta) = 3 \theta - \sin \theta - 2 \pi$
$\displaystyle f'(\theta) = 3 - cos \theta >0$
So $\displaystyle f(\theta)$ is increasing.

$\displaystyle f(0) = -2\pi < 0$
$\displaystyle f(\pi) = \pi > 0$
so there is a root in $\displaystyle [0,\pi]$
so on....

3. is that just pretty much using guess and check? i was hoping for an exact answer.
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edit
it seems that if i was to use that method i'd just use a calculator (not meaning to sound abrasive)

4. Originally Posted by dinNA89
is that just pretty much using guess and check? i was hoping for an exact answer.
---------------
edit
it seems that if i was to use that method i'd just use a calculator (not meaning to sound abrasive)
well...thts as much as i can help...

5. i don't want you to feel unappreciated, i think it's admirable that you (and so many others) find time to help strangers with maths.

6. Originally Posted by dinNA89
i don't want you to feel unappreciated, i think it's admirable that you (and so many others) find time to help strangers with maths.
absolutely not...and reading threads here is my way to learn the subject...

7. I believe Newton Method would have to be used or some other iterative method. It may be able to be solved analytically, but I kind of doubt it.

8. Originally Posted by dinNA89
I wasn't sure where to post this as i'm not sure if it's blindingly simple or more difficult, I'm an out of practice engineering student so either is just as likely
anyway, just trying to solve for theta
$\displaystyle 3 \theta - \sin \theta = 2 \pi$

I could easily get an answer using a calculator, but i want to do it analytically.

you can't find the exact value of it, you just can solve it numerically by bisection method or newtons method

see this link for newton's method

Newton's Method

newton's method converges faster than bisection method so it is better

9. Originally Posted by Amer
you can't find the exact value of it, you just can solve it numerically by bisection method or newtons method

see this link for newton's method

Newton's Method

newton's method converges faster than bisection method so it is better

Newton's Method is more computationally expensive though, so I would not necessarily call it better. Secant method is another iteration method that can be used. Also, these methods will converge to different values based on an initial guess, because you are finding a root to a function that has more than one root. The root that you find will be dependent on the initial guess, so make sure you guess something near where you expect the solution to be based on your own requirements.

10. yeah, i'm familiar with both of those methods and was hoping not to have to use them. Is there absolutely no way to get an exact solution?

11. can you prove that its root is a rational or irrational number ?

12. Originally Posted by dinNA89
yeah, i'm familiar with both of those methods and was hoping not to have to use them. Is there absolutely no way to get an exact solution?
Probably not.
If you could get something exact, then you would have a method of calculating $\displaystyle \pi$

$\displaystyle \pi = \dfrac{3 \theta - \sin \theta }{ 2}$

Seems to be an iteration problem
.