Question

find coterminal, reference, and quadrant
question
for the rotation $-783^\circ$, 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 \\(\square\\), with a reference angle of \\(\square^\circ\\).

Explanation:

Step1: Find coterminal angle

To find a coterminal angle of \(-783^\circ\) within \(0^\circ \leq \theta < 360^\circ\), we add multiples of \(360^\circ\) until we get an angle in that range. Let's find how many times we need to add \(360^\circ\).

Let \(n\) be the number of times we add \(360^\circ\). We solve \(-783 + 360n \geq 0\).

\(360n \geq 783\)

\(n \geq \frac{783}{360} \approx 2.175\). So we take \(n = 3\).

Now calculate \(-783 + 360\times3=-783 + 1080 = 297^\circ\).

Step2: Determine the quadrant

Angles: \(0^\circ - 90^\circ\) (Quadrant I), \(90^\circ - 180^\circ\) (Quadrant II), \(180^\circ - 270^\circ\) (Quadrant III), \(270^\circ - 360^\circ\) (Quadrant IV).

\(270^\circ < 297^\circ < 360^\circ\), so it lies in Quadrant IV.

Step3: Find reference angle

For an angle in Quadrant IV (\(270^\circ < \theta < 360^\circ\)), the reference angle is \(360^\circ - \theta\).

So reference angle \(= 360 - 297 = 63^\circ\).

Answer:

The coterminal angle is \(297^\circ\), which lies in Quadrant IV, with a reference angle of \(63^\circ\).