Example  4
Portia is playing a game of around the world darts. In the game, she must hit each number, in order, from \(1\) to \(20\).

  1. What angle does each number’s wedge make with the bull’s eye (centre of the dart board), if there are \(20\) wedges?

    This can be solved using a proportion.

    \[\begin{align}
     \frac{1}{{20}} &= \frac{x}{{360^\circ }} \\
     x &= 18^\circ  \\
     \end{align}\]

    Each wedge makes an angle of \(18^\circ\) with the centre of the dart board.

  2. Determine the number whose wedge, in standard position, ranges from \(99^\circ\) to \(117^\circ\).

    The number \(20\)’s wedge will range from \(90^\circ - 9^\circ = 81^\circ\) to \(90^\circ + 9^\circ = 99^\circ\), therefore the desired number will be next to \(20\) in Quadrant II, which is the number \(5\).

  3. What other numbers have the same reference angle range as the number from part b.?

    The reference angle range of the number \(5\) is:

    \(\begin{align}
     \theta _R &= 180^\circ - 117^\circ \\
     &= 63^\circ \end{align}\)

    to

    \(\begin{align}
     \theta _R &= 180^\circ - 99^\circ \\
     &= 81^\circ \end{align}\)

    Numbers with the same reference angle range can be found by reflecting these reference angles across the \(y\)-axis and \(x\)-axis.

    Reflecting across the \(y\)-axis will give the number \(1\). Reflecting the number \(5\) and the number \(1\) across the \(x\)-axis will give \(19\) and \(17\), respectively.