finding the acute angle between two functions at first point of intersection

Question

I need to determine the acute angle between two curves at the first two points of intersection, both graphically and algebraically. The functions for the curves are: sin(0.5x) and cos(2x+2), a picture of the functions graphed are attached below. I have been stuck on this problem for awhile now and would greatly appreciate any solutions. functions graphed, Point A represents first point of intersection

Answer 1

Intersection point is $A$ is where $\sin 0.5x=\cos(2x+2)$

That is $x_A\approx 1.80826$

Derivative of $g(x)=\cos(2x+2)$ is $g'(x)=-2 \sin (2 x+2)$

Slope of tangent is $m=g'(x_A)=1.23674$

Derivative of $f(x)=\sin 0.5x$ is $f'(x)=\dfrac{1}{2} \cos \dfrac{x}{2}$

slope of tangent is $m'=f'(x_A)=0.309185$

The angle formed by the two tangent line is $\alpha$ and

$\tan \alpha=\dfrac{m-m'}{1+mm'}\approx 0.75$

$\alpha=\arctan 0.75\approx 36.86°$