Question

question
for the rotation $-401^\circ$, find the coterminal angle from $0^\circ \leq \theta < 360^\circ$, the quadrant, and the reference angle.
answer
attempt 1 out of 2
the coterminal angle is $\square^\circ$, which lies in quadrant $\downarrow$, with a reference angle of $\square^\circ$.

Explanation:

Step1: Find coterminal angle

To find a coterminal angle with \(-401^\circ\) in the range \(0^\circ\leq\theta < 360^\circ\), we add multiples of \(360^\circ\) until we get an angle in that range. Let's add \(360^\circ\) twice (since adding once gives \(-401 + 360=-41^\circ\), still negative). So, \(-401^\circ+ 2\times360^\circ=-401^\circ + 720^\circ = 319^\circ\).

Step2: Determine the quadrant

Angles between \(270^\circ\) and \(360^\circ\) lie in Quadrant IV. Since \(270^\circ<319^\circ < 360^\circ\), the angle \(319^\circ\) is in Quadrant IV.

Step3: Find the reference angle

For an angle in Quadrant IV (\(270^\circ<\theta<360^\circ\)), the reference angle is given by \(360^\circ-\theta\). So, reference angle \(= 360^\circ - 319^\circ=41^\circ\).

Answer:

The coterminal angle is \(319^\circ\), which lies in Quadrant IV, with a reference angle of \(41^\circ\).