How to solve this? (Argument of the complex number in complex plane)

Let $z=r(\cos\theta+i\sin\theta).$

Thus, $$r^2\cos^2\theta+(r\sin\theta-10)^2=36$$ or $$\sin\theta=\frac{r^2+64}{20r}\geq\frac{2\sqrt{64r^2}}{20r}=\frac{4}{5},$$ which gives $$\arcsin\frac{4}{5}\leq\theta\leq\pi-\arcsin\frac{4}{5}.$$ Also, write $$8\sin\theta+6\cos\theta=10\sin\left(\theta+\arccos\frac{4}{5}\right).$$ Can you end it now?

Actually, I assumed that $0\leq\arg{z}<2\pi.$

I got $$2.8\leq8\sin\theta+6\cos\theta\leq10.$$

I used the following reasoning.

By C-S $$8\sin\theta+6\cos\theta\leq\sqrt{8^2+6^2)(\sin^2\theta+\cos^2\theta)}=10.$$ The equality occurs for $(\sin\theta,\cos\theta)||(4,3)$, which gives $\theta=\arctan\frac{4}{3}\in\left[\arcsin\frac{4}{5},\pi-\arcsin\frac{4}{5}\right]$, which says that we got a maximal value.

Also, $$f(x)=8\sin{x}+6\cos{x}=10\sin\left(x+\arccos\frac{4}{5}\right)$$ is a concave function on $\left[\arcsin\frac{4}{5},\pi-\arcsin\frac{4}{5}\right],$ which says $$\min\limits_{\left[\arcsin\frac{4}{5},\pi-\arcsin\frac{4}{5}\right]}f=\min\left\{f\left(\arcsin\frac{4}{5}\right),f\left(\pi-\arcsin\frac{4}{5}\right)\right\}=f\left(\pi-\arcsin\frac{4}{5}\right)=2.8.$$

If you draw a diagram, then the equation describes a circle of radius $6$ centred at $10$ units above the origin. Thus, we have that $r\cos\theta$ ranges from $-6$ to $6$ and that $r\sin\theta$ ranges between the extremes of $4$ and $16,$ with $r=|z|$ ranging between $4$ and $16.$

Now if we set $\cos\theta=x$ and $\sin\theta=y,$ then it follows that the maximum and minimum of $x$ occur at one of the endpoints of $[-6/r,6/r],$ with $r$ being either of the extremes in turn, which can be checked manually. Do the same thing for $y.$ Then the problem is now that of finding the extreme values of $6x+8y$ over a certain rectangle defined by the extreme values of $x$ and $y$ calculated above. For example, we have that $x$ varies between $-3/2$ and $3/2.$ Similarly, you can work out the range of values of $y,$ and the extremes of the linear function $6x+8y$ will now occur at the endpoints of the intervals calculated.