What does \(\arcsin x\) mean?

State the principal range of \(\arccos x\).

Evaluate \(\arcsin\left(\frac12\right)\).

Evaluate \(\arctan(1)\).

Evaluate \(\arccos\left(-\frac12\right)\).

Find \(\sin(\arccos x)\) for \(-1\le x\le1\).

Solve \(\sin\theta=\frac{\sqrt2}{2}\) for the principal value \(\theta=\arcsin\left(\frac{\sqrt2}{2}\right)\).

Evaluate \(\arcsin(\sin(\frac{5\pi}{6}))\).

Find \(\cos(\arcsin x)\) for \(-1\le x\le1\).

Find \(\tan(\arcsin x)\) for \(-1<x<1\).

Explain why \(\arcsin(2)\) is not a real number.

Show that \(\arcsin x+\arccos x=\frac\pi2\) for \(-1\le x\le1\).

Find the angle \(\theta\) in \((-\frac\pi2,\frac\pi2)\) whose tangent is \(-\sqrt3\).

Find \(\arctan\left(\frac{y}{x}\right)\) for \(x=3\), \(y=3\), and state what angle it gives.

Solve \(\arccos x=\frac{2\pi}{3}\).

Find all real \(x\) for which \(\arcsin x\) and \(\arccos x\) are both defined.

A student says \(\sin^{-1}x=\frac1{\sin x}\). Explain the notation error in the context of inverse trig functions.

Prove that \(\tan(\arctan x)=x\) for every real \(x\), and explain why the reverse \(\arctan(\tan\theta)=\theta\) needs a restriction.

Evaluate \(\arccos(\cos(\frac{5\pi}{3}))\) and explain the principal-value issue.

A vector has components \((-1,1)\). Explain why \(\arctan(y/x)\) alone does not identify its quadrant, and find the correct direction angle in \([0,2\pi)\).