finding the acute angle between two functions at first point of intersection
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
$\tan \alpha=\dfrac{mm'}{1+mm'}\approx 0.75$
$\alpha=\arctan 0.75\approx 36.86°$
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Updated on December 24, 2022Comments

Admin 11 months
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