Solving an equation

Question

Showing $\sqrt a + $$\sqrt {\cos(\sin a)} = 2$

I've attempted various manipulations (multiplying by one, squaring, etc.) but cannot find a way to solve for a. Anyone have an idea how I can approach this problem? Thanks.

Answer 1

It isn't actually possible to solve for $a$, but we can do some simple fixed-point iteration:

$$\sqrt a=2-\sqrt{\cos(\sin a)}\implies a=\left(2-\sqrt{\cos(\sin a)}\right)^2$$

We rewrite this as

$$a_{n+1}=\left(2-\sqrt{\cos(\sin a_n)}\right)^2$$

And start off with a guess $a_0=1$.

$$a_1=\left(2-\sqrt{\cos(\sin a_0)}\right)^2=1.401115158$$

$$a_2=\left(2-\sqrt{\cos(\sin a_1)}\right)^2=1.579572306$$

$$a_3=\left(2-\sqrt{\cos(\sin a_2)}\right)^2=1.600036196$$

$$a_4=\left(2-\sqrt{\cos(\sin a_3)}\right)^2=1.599473216$$

And this sequence will approach the solution. I would personally use this method if you lack an understanding of derivatives. If you did have an understanding of derivatives, Newton's method would most likely work much better.

Answer 2

The solution is approximately $a = 1.5994958620742425268$, but is very unlikely to be expressible in closed form. Numerical methods (e.g. Newton's method) work well.

Answer 3

For this kind of equation, there is no analytical solution (it is already the case for $x=\cos(x)$) and numerical methods should be used.

As Robert Israel already answered, Newton method could be the simplest to use considering $$f(x)=\sqrt{x}+\sqrt{\cos (\sin (x))}-2$$ Starting from a "reasonable" guess $x_0$, the method will updtae it according to $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$ For the present case $$f'(x)=\frac{1}{2 \sqrt{x}}-\frac{\sin (\sin (x)) \cos (x)}{2 \sqrt{\cos (\sin (x))}}$$ By inspection, we know that the root is between $0$ and $\pi$ since $f(0)=-1$ and $f(\pi)=\sqrt{\pi }-1$. So, let us start at the mid point $x_0=\frac \pi 2$.

Newton iterates will then be the following $$\left( \begin{array}{cc} n & x_n \\ 0 & 1.5707963267948966192 \\ 1 & 1.5999566241894927879 \\ 2 & 1.5994959773991264235 \\ 3 & 1.5994958620742497549 \\ 4 & 1.5994958620742425268 \end{array} \right)$$ which the solution for twenty significant figures.