求 $3\cos\dfrac{2\pi}5-\cos\dfrac{\pi}5$ 的值．
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【答案】
$\dfrac{-2+\sqrt 5}2$
【解析】
由\[\cos\dfrac{\pi}5+\cos\dfrac{3\pi}5+\cos\dfrac{5\pi}5+\cos\dfrac{7\pi}5+\cos\dfrac{9\pi}5=0,\]可得\[\cos\dfrac{\pi}5-\cos\dfrac{2\pi}5=\dfrac 12.\]又\[\cos\dfrac{\pi}5\cdot\cos\dfrac{2\pi}5=\dfrac{\sin\dfrac{\pi}5\cos\dfrac{\pi}5\cos\dfrac{2\pi}5}{\sin\dfrac{\pi}5}=\dfrac 14.\]于是\[\begin{split}3\cos\dfrac{2\pi}5-\cos\dfrac{\pi}5&=-2\left(\cos\dfrac{\pi}5-\cos\dfrac{2\pi}5\right)+\left(\cos\dfrac{\pi}5+\cos\dfrac{2\pi}5\right)\\
&=-1+\sqrt{\left(\dfrac 12\right)^2+4\cdot \dfrac 14}\\
&=\dfrac{-2+\sqrt 5}2.\end{split}\]
&=-1+\sqrt{\left(\dfrac 12\right)^2+4\cdot \dfrac 14}\\
&=\dfrac{-2+\sqrt 5}2.\end{split}\]
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