If \(\theta\) is an angle in a quadrant such that \(\theta \gt 360^\circ\), then the position of \(\theta\) can be determined by subtracting a multiple of \(360^\circ\) or \(2\pi\) rad to obtain an angle that corresponds to \(0^\circ \leq \theta \leq 360^\circ\) or \(0 \leq \theta \leq 2\pi\) rad.
Determine the position of each of the following angles in the quadrants:
(a) \(800^\circ\) (b) \(\dfrac{19}{6}\pi \text{ rad}\)
(a)
\(\begin{aligned} 800^{\circ} - 2(360^{\circ}) &= 80^{\circ}\\ 800^{\circ} &=2(360^{\circ}) +80^{\circ}.\\ \end{aligned}\)
\(80^\circ\) is the corresponding angle of \(800^\circ\).
Since \(80^\circ\) lies in Quadrant \(\text{I}\),
thus \(800^\circ\) lies in Quadrant \(\text{I}\).
(b)
\(\begin{aligned} \dfrac{19}{6} \pi \text{ rad} - 2\pi \text{ rad} &= \dfrac{7}{6}\pi \text{ rad}\\ \dfrac{19}{6} \pi \text{ rad} &= 2\pi \text{ rad} + \dfrac{7}{6}\pi \text{ rad}. \end{aligned}\)
\(\dfrac{7}{6}\pi\) rad is the corresponding angle of \(\dfrac{19}{6}\pi\) rad.
Since \(\dfrac{7}{6}\pi\) rad lies in Quadrant \(\text{III}\),
thus \(\dfrac{19}{6}\pi\) rad lies in Quadrant \(\text{III}\).
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