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 :) ).
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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. |
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if $\cos x=2x$ which means $$x=\frac {1}{2.22131587} \,\mathrm{rad},$$ $$x=28.65957881\,\mathrm{Grad},$$ $$x=25.79362093\,\mathrm{Degree}$$ 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!. $$\cos x \simeq 1-\frac {x^2}{2!}=2x,$$ $$x^2+4x-2=0,$$ $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ where the $a=1$ and $b=4$ and $c=-2$ then $x\simeq 0.45$ |
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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. It's so simple. |
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