1. To graph \(\alpha=60^{\circ}\), we draw an angle with its initial side on the positive \(x\)-axis and rotate counter-clockwise \(\frac{60^{\circ}}{360^{\circ}}=\frac{1}{6}\) of a revolution. We see that \(\alpha\) is a Quadrant I angle.

To find angles which are coterminal, we look for angles \(\theta\) of the form \(\theta=\alpha+360^{\circ} \cdot k\) for some integer \(k\).

- When \(k=1\), we get \(\theta=60^{\circ}+360^{\circ}=420^{\circ}\).

- Substituting \(k=-1\) gives \(\theta=60^{\circ}-360^{\circ}=-300^{\circ}\).

- If we let \(k=2\), we get \(\theta=60^{\circ}+720^{\circ}=780^{\circ}\).

2. Since \(\beta=-225\) is negative, we start at the positive \(x\)-axis and rotate clockwise \(\frac{225}{360^{\circ}}=\frac{5}{8}\) of a revolution. We see that \(\beta\) is a Quadrant II angle.

To find coterminal angles, we proceed as before and compute \(\theta=-225^{\circ}+360^{\circ} \cdot k\) for integer values of \(k\). Letting \(k=1\)

\(k=-1\) and \(k=2\), we find \(135^{\circ},-585^{\circ}\) and \(495^{\circ}\) are all coterminal with \(-225^{\circ}\).

3. Since \(\gamma=540\) is positive, we rotate counter-clockwise from the positive \(x\)-axis. One full revolution accounts for \(360^{\circ}\), with \(180^{\circ}\), or half of a revolution, remaining. Since the terminal side of \(\gamma\) lies on the negative \(x\)-axis, \(\gamma\) is a quadrantal angle.

All angles coterminal with \(\gamma\) are of the form \(\theta=540^{\circ}+360^{\circ} \cdot k\), where \(k\) is an integer. Working through the arithmetic, we find three such angles: \(900^{\circ}, 180^{\circ}\) and \(-180^{\circ}\).

4. The Greek letter \(\varphi\) is pronounced 'fee' or 'fie'9 and since \(\varphi=-750 \quad\) is negative, we begin our rotation clockwise from the positive \(x\)-axis. Two full rotations account for \(720^{\circ}\), with just \(30^{\circ}\) or \(\frac{1}{12}\) of a revolution to go.

We find that \(\varphi\) is a Quadrant IV angle. To find coterminal angles, we compute \(\theta=-750^{\circ}+360^{\circ} \cdot k\) for a few integers \(k\) and obtain \(-390^{\circ},-30^{\circ}\) and \(330^{\circ} .\)