9. Given the sequence: \( 1 ; \sin 45^{\circ} ; \frac{1}{2} ; \ldots \) a Write down the next two terms if the pattern continue b Find the \( n \)th term.
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Bonus Knowledge
The sequence you provided starts with \( 1, \sin 45^{\circ}, \frac{1}{2}, \ldots \). The next two terms can be derived using the sine function with increments of angles. The angle increments appear to be \( 0^{\circ}, 45^{\circ}, 30^{\circ} \) (noticing from \(\sin 45^{\circ} = \frac{\sqrt{2}}{2} \approx 0.707\) and \(\frac{1}{2} = \sin 30^{\circ}\)). The next two angles should logically be \( 60^{\circ} \) and \( 75^{\circ} \). a. The next two terms will be \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \) and \( \sin 75^{\circ} = \frac{\sqrt{6} + \sqrt{2}}{4} \). b. The \( n \)th term can be described as \( \sin\left(45^{\circ + 15(n-1)}\right) \) for \( n \geq 1\). This indicates the increments of \( 15^{\circ} \) for each step following the initial \( 45^{\circ} \) term.