Preview Activity 2.2.1.
In the following figure there are 24 equally spaced points on the unit circle. Since the circumference of the unit circle is \(2\pi\text{,}\) each of the points is \(\frac{1}{24} \cdot 2\pi = \frac{\pi}{12}\) units apart (traveled along the circle). Thus, the first point counterclockwise from \((1,0)\) corresponds to the distance \(t = \frac{\pi}{12}\) traveled along the unit circle. The second point is twice as far, and thus \(t = 2 \cdot \frac{\pi}{12} = \frac{\pi}{6}\) units along the circle away from \((1,0)\text{.}\)
(a)
Label each of the subsequent points on the unit circle with the exact distance they lie counter-clockwise away from \((1,0)\text{;}\) write each fraction in lowest terms.
(b)
Which distance along the unit circle corresponds to \(\frac{1}{4}\) of a full rotation around? to \(\frac{5}{8}\) of a full rotation?
(c)
One way to measure angles is connected to the arc length along a circle. For an angle whose vertex is at \((0,0)\) in the unit circle, we say the angle’s measure is \(1\) radian provided that the angle intercepts an arc of the circle that is \(1\) unit in length, as pictured in the following figure. Note particularly that an angle measuring \(1\) radian intercepts an arc of the same length as the circle’s radius.
Suppose that \(\alpha\) and \(\beta\) are angles with respective radian measures \(\alpha = \frac{\pi}{3}\) and \(\beta = \frac{3\pi}{4}\text{.}\) Assuming that we view \(\alpha\) and \(\beta\) as having their vertex at \((0,0)\) and one side along the positive \(x\)-axis, sketch the angles \(\alpha\) and \(\beta\) on the unit circle in part (a).
(d)
What is the radian measure that corresponds to a \(90^\circ\) angle?
