<s>[INST] Solve algebraically and graphically: $\arcsin(x) + \arccos\left(\frac{x}{2}\right) = \frac{5\pi}{6}$ Solve:
$$\arcsin(x) + \arccos\left(\frac{x}{2}\right) = \frac{5\pi}{6}$$
I think the algebraic solution should start like:
$$\arcsin(x) = \frac{5\pi}{6} - \arccos\left(\frac{x}{2}\right)$$
$$x = \sin\left(\frac{5\pi}{6} - \arccos\left(\frac{x}{2}\right)\right)$$
at that stage probably I should use some trigonometric relation or property of $\arccos(x)$ to convert it to $\arcsin(x)$, however I can't figure it out.
As for the solution by graph I can't even think what steps should I follow to build it.
 [/INST] $$\arcsin x + \arccos\frac{x}{2} = \frac{5\pi}{6}$$
$$\arcsin x = \frac{5\pi}{6} - \arccos\frac{x}{2}$$
$$x = \sin{\left(\frac{5\pi}{6} - \arccos\frac{x}{2}\right)}$$
$$x=\sin\frac{5\pi}6\cos \left(\arccos\frac{x}{2}\right)-\cos\frac{5\pi}6 \sin\left(\arccos\frac{x}{2}\right)$$
$$x=\frac12\cdot\frac x2+\frac{\sqrt3}{2} \cdot\sqrt{1-\left(\frac x2\right)^2}$$
Answer: $x=1$
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