Question

determine the reference angle ( \theta ).

( \theta )	quadrant	+ coterminal angle	ref. angle ( \theta )
( 480^{circ} )	( qii )	( 120^{circ} )	( 60^{circ} )
( \frac{15pi}{7} )	( q1 )	( \frac{15pi}{7} )	( \frac{pi}{7} )
( -\frac{pi}{8} )	( qiv )	( \frac{7pi}{4} )	( \frac{pi}{4} )

Explanation:

Step1: Determine the quadrant for \(-135^{\circ}\)

Since \(-135^{\circ}+360^{\circ} = 225^{\circ}\), and \(180^{\circ}<225^{\circ}<270^{\circ}\), the quadrant is \(QIII\) (not \(Q1\) as wrongly written before).

Step2: Determine the coterminal angle for \(480^{\circ}\)

\(480^{\circ}-360^{\circ}=120^{\circ}\) (not \(170^{\circ}\) as wrongly written before).

Step3: Determine the quadrant for \(\frac{15\pi}{7}\)

\(\frac{15\pi}{7}-2\pi=\frac{15\pi - 14\pi}{7}=\frac{\pi}{7}\), and \(0<\frac{\pi}{7}<\frac{\pi}{2}\), the quadrant is \(Q1\) (correct).

Step4: Determine the coterminal angle for \(-\frac{\pi}{8}\)

\(-\frac{\pi}{8}+2\pi=\frac{- \pi+16\pi}{8}=\frac{15\pi}{8}\) (not \(\frac{7\pi}{4}\) as wrongly written before).

Answer:

For \(\theta=-135^{\circ}\): Quadrant \(QIII\), Coterminal Angle \(225^{\circ}\), Ref. Angle \(45^{\circ}\)
For \(\theta = 480^{\circ}\): Quadrant \(QII\), Coterminal Angle \(120^{\circ}\), Ref. Angle \(60^{\circ}\)
For \(\theta=\frac{15\pi}{7}\): Quadrant \(Q1\), Coterminal Angle \(\frac{\pi}{7}\), Ref. Angle \(\frac{\pi}{7}\)
For \(\theta=-\frac{\pi}{8}\): Quadrant \(QIV\), Coterminal Angle \(\frac{15\pi}{8}\), Ref. Angle \(\frac{\pi}{8}\)