3.
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\caption{Figure 1}
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Figure 1 shows a regular pentagon \(O A B C D\). The vectors \(\mathbf { p }\) and \(\mathbf { q }\) are defined by \(\mathbf { p } = \overrightarrow { O A }\) and \(\mathbf { q } = \overrightarrow { O D }\) respectively.
Let \(k\) be the number such that \(\overrightarrow { D B } = k \overrightarrow { O A }\).
- Write down \(\overrightarrow { A C }\) in terms of \(\mathbf { p } , \mathbf { q }\) and \(k\) as appropriate.
- Show that \(\overrightarrow { C D } = - \mathbf { p } - \frac { 1 } { k } \mathbf { q }\)
- Hence find the value of \(k\)
By considering triangle \(D B C\), or otherwise,
- find the exact value of \(\sin 54 ^ { \circ }\)