Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I would like to get an approx. solution to the equation: $ \cos x = 2x$, I don't need an exact solution just some approx. And I need the solution without higher mathematics (without derive and things like that :) ).

share|cite|improve this question
@HansLundmark I need to present some solution not just giving some numbers from some computer. – gen Jan 10 '13 at 15:33
Hint: Do a Taylor series expansion of $\cos x$ (say, up to order 6) on the left and then solve for x in the remaining equation. Now, explain this approach and figure out how to select the right value and what the error is. Regards – Amzoti Jan 10 '13 at 15:49
try using the fact that $ |\cos x| \le 1 $ and $ \cos (-x) = \cos (x)$ – Santosh Linkha Jan 10 '13 at 15:49
@gen: The word "solution" is ambiguous. It can mean both the value of $x$ which satisfies the equation, and the procedure for finding that value. When you say "an approx. solution", I think most people will interpret this as just the number, so if you want the procedure you should perhaps have said so in the question. – Hans Lundmark Jan 10 '13 at 18:30
up vote 15 down vote accepted

Take a pocket calculator, start with $0$ and repeatedly type [cos], [$\div$], [2], [=]. This will more or less quickly converge to a value $x$ such that $\frac{\cos x}2=x$, just what you want.

share|cite|improve this answer
Thank you it will be fine :) – gen Jan 10 '13 at 16:44
Or [$\cos$], [$2$], [$/$] if you are using an RPN calculator. – robjohn Jan 10 '13 at 16:48
@robjohn just imagine if this question was deleted!. – Neo Jan 10 '13 at 17:20
Make sure you've got it set on RADIANS, not DEGREES. – Michael Hardy Jan 10 '13 at 17:51

At small angles $\alpha ≈ sin\alpha$, so at the equation we can write $2 sin\alpha≈ cos \alpha$. From that $tan \alpha ≈ 1/2$. The solution to the last equation is less than a degree off the exact solution to the original equation.

share|cite|improve this answer

if $\cos x=2x$ which means $$x=\frac {1}{2.22131587} \,\mathrm{rad},$$



point: $$\cos x=1-\frac {x^2}{2!}+\frac {x^4}{4!}-\frac {x^6}{6!}+\cdots$$

$0 < x < +\frac {1}{2.5}\,\mathrm{rad}$ because $\cos x=2x$

This is also a mathematical method, i mean Estimate the amount through math without using a computer or calculator!.

$$\cos x \simeq 1-\frac {x^2}{2!}=2x,$$



where the $a=1$ and $b=4$ and $c=-2$ then $x\simeq 0.45$

share|cite|improve this answer
I need some reasing... – gen Jan 10 '13 at 15:47
What method did you use to get the approximate solution? – robjohn Jan 10 '13 at 16:16
I've written a complex computer program for solving these equations. – Neo Jan 10 '13 at 16:24
How is this supposed to help? The two upvoters might want to explain, as well. – Did Jan 10 '13 at 16:26
Neo: Sorry? What is based on what? – Did Jan 10 '13 at 20:34

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.