$$ -\frac{\pi}{3} $$ Because the range of arcsine is \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\), the answer must be given as \(-\frac{\pi}{3}\), not \(\frac{5\pi}{3}\).
$$ \frac{\pi}{4} $$ Within \([0, \pi]\), \(\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\).
Evaluate: $$ \arccos\left(-\frac{1}{2}\right) $$
$$ \frac{2\pi}{3} $$ Negative inputs to arccosine yield angles in Quadrant II: \(\pi - \frac{\pi}{3} = \frac{2\pi}{3}\).
Why does \(\arccos\left(-\frac{\sqrt{3}}{2}\right)\) equal \(\frac{5\pi}{6}\) rather than \(-\frac{\pi}{6}\)?
The range of \(\arccos(x)\) is \([0, \pi]\). Angles in Quadrant IV such as \(-\frac{\pi}{6}\) fall entirely outside this defined range.
Evaluate: $$ \arctan(1) $$
$$ \frac{\pi}{4} $$ In \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), the unique angle with \(\tan(\theta) = 1\) is \(\frac{\pi}{4}\).
Evaluate: $$ \arctan(-\sqrt{3}) $$
$$ -\frac{\pi}{3} $$ Because the range of arctangent is \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), negative inputs yield negative Quadrant IV angles.
Evaluate: $$ \arcsin(0) $$
$$ 0 $$ \(\sin(0) = 0\), and \(0\) lies within \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\).
Evaluate: $$ \arccos(-1) $$
$$ \pi $$ \(\cos(\pi) = -1\), and \(\pi\) lies within \([0, \pi]\).
Evaluate: $$ \arctan(0) $$
$$ 0 $$ \(\tan(0) = 0\), and \(0\) lies within \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\).
Why is the expression undefined in the real numbers? $$ \arcsin(2) $$
The domain of \(\arcsin(x)\) is \([-1, 1]\). Because \(\sin(\theta)\) can never exceed \(1\), no real angle \(\theta\) satisfies \(\sin(\theta) = 2\).
How does solving \(\sin(\theta) = -\frac{1}{2}\) on \([0, 2\pi)\) differ from evaluating \(\arcsin\left(-\frac{1}{2}\right)\)?
Solving the equation yields two solutions in \([0, 2\pi)\): \(\theta = \frac{7\pi}{6}\) and \(\theta = \frac{11\pi}{6}\). The function evaluation \(\arcsin\left(-\frac{1}{2}\right)\) has only one principal output: \(-\frac{\pi}{6}\).
Evaluate: $$ \arccos(0) $$
$$ \frac{\pi}{2} $$ \(\cos\left(\frac{\pi}{2}\right) = 0\), which is the unique angle in \([0, \pi]\).
