IB Mathematics AI AHL area of sector, length of arc MAI Study Notes - New Syllabus

The values of the angles can be represented well on the below trigonometric circle:

In fact, each value on the circle indicates the angle between the corresponding radius and the positive x-axis radius (red arrow). The angle formed after a complete circle is \(360^\circ\). The angle formed after half a circle is \(180^\circ\). However, after completing a full circle (\(1^\text{st}\) period) we can continue counting:

$361^\circ, 362^\circ, 263^\circ \text{ and so on}$

The next full circle (\(2^\text{nd}\) period) finishes at \(2 \times 360^\circ = 720^\circ\). Similarly, we can move clockwise, considering negative angles:

$-1^\circ, -2^\circ, -3^\circ \text{ and so on}$

For example, \(270^\circ\) can also be seen as \(-90^\circ\). Therefore, an angle may have any value from \(-\infty\) to \(+\infty\).

Consider the point on the unit circle corresponding to \(30^\circ\) or \(\frac{\pi}{6}\) radians.

Let’s start from \(0^\circ\) and move anticlockwise. We pass through \(30^\circ\) and after completing a full circle, we pass through the same point at \(30^\circ + 360^\circ = 390^\circ\) and then again at \(30^\circ + 360^\circ \times 2 = 750^\circ\), and so on. In other words, we add (or subtract) multiples of \(360^\circ\):

$30^\circ + 360^\circ k \quad \text{where } k \in \mathbb{Z}$

Thus, for \(k = \ldots, -1, 0, 1, 2, \ldots\), we obtain the values:

$\ldots, -330^\circ, 30^\circ, 390^\circ, 750^\circ, \ldots \quad \text{[in degrees]}$

Similarly, in radians, we add multiples of \(2\pi\):

$\frac{\pi}{6} + 2k\pi \quad \text{where } k \in \mathbb{Z}$

Thus, the point has infinitely many angle values:

$\ldots, -\frac{11\pi}{6}, \frac{\pi}{6}, \frac{13\pi}{6}, \frac{25\pi}{6}, \ldots \quad \text{[in radians]}$

NOTE:

If we consider only the positive values of the angles, we have an arithmetic sequence with \(u_1 = 30^\circ\) and \(d = 360^\circ\) (or \(u_1 = \frac{\pi}{6}\) and \(d = 2\pi\) in radians).