How can I find the points of intersection of $f(x)=\sin(\sqrt{x^2+4})$ and $g(x)=4-x^2$ analytically How can I find the points of intersection of $f(x)=\sin(\sqrt{x^2+4})$ and $g(x)=4-x^2$ analytically without having to resort to a graph? Thank you very much
I know that $-1\leq\sin(\sqrt{x^2+4})\leq 1$, with which $-1\leq4-x^2\leq 1$ and $-1\leq (2-x)(2+x)\leq 1$, but I do not know what else to do, could you help me?
 A: HINT: write your equation in the form
$$\sin\left(\sqrt{x^2+4}\right)-(4-x^2)=0$$ and use a numerical method.
this can not be done analytically!
$$x\approx -1.90515677885686969692$$
$$x\approx 1.90515677885686969692$$
A: Let $u=\sqrt{x^2+4}$.  Then $x^2=u^2-4$, so the equation to solve becomes
$$\sin u=8-u^2$$
with $u$ understood to be positive.  Since $|\sin u|\le1$, there is a solution $u\in[\sqrt7,3]$, which corresponds to $|x|\in[\sqrt3,\sqrt5]$.  But the equation has no nice closed-form solution; your best bet is a numerical approximation.
One way to get an approximation is by iterating the recursion
$$u_{n+1}=\sqrt{8-\sin u_n}$$
starting say with $u_n=3$.  This leads to $u\approx2.762177$, or $x\approx1.90516$.
A: Just to add a few things after Jam's answer.
Sooner or later, you will learn than, better than with Taylor expansions, functions can be approximated using Padé approximants which, built around $u=u_0$, are in the form of $$f(u)=\frac{\sum_{i=0}^m a_i (u-u_0)^i } {1+\sum_{i=1}^n b_i (u-u_0)^i}$$ Using, for simplicity, $m=1$, this gives as an approximate solution $$u_{(n)}=u_0-\frac{a_0^{(n)}}{a_1^{(n)}}$$ (remember that $a_0$ and $a_1$ depend on the degree $n$ used for the denominator).
Applying the method to $f(u)=\sin(u)+u^2-8$ with $u_0=\pi$, we could obtain
$$\left(
\begin{array}{ccc}
  n & u_{(n)} & \text{approx} \\
 0 & \frac{8-\pi +\pi ^2}{2 \pi -1} & 2.78771 \\
 1 & \frac{-8+25 \pi -3 \pi ^2+\pi ^3}{9-4 \pi +3 \pi ^2} & 2.76231 \\
 2 & \frac{432-230 \pi +318 \pi ^2-52 \pi ^3+6 \pi ^4+\pi ^5}{-38+228 \pi -76 \pi
   ^2+24 \pi ^3+\pi ^4} & 2.76208 \\
 3 & \frac{-1824+13380 \pi -5628 \pi ^2+3612 \pi ^3-600 \pi ^4+24 \pi ^5+18 \pi
   ^6}{2436-2208 \pi +3828 \pi ^2-1104 \pi ^3+168 \pi ^4+24 \pi ^5} & 2.76218
\end{array}
\right)$$
