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

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

$\alpha=\arctan 0.75\approx 36.86°$

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Updated on December 24, 2022