Question

Determine the value of 3\cos\dfrac{2\pi}{5}-\cos\dfrac{\pi}{5}

Collected in the board: Trigonometry

Steven Zheng Steven Zheng posted 6 days ago


Answer

\cos \dfrac{\pi }{5}=\dfrac{1+\sqrt{5} }{4}

\dfrac{2\pi }{5} =2\cos^2\dfrac{\pi}{5} -1

=2\cdotp (\dfrac{1+\sqrt{5} }{4} )^2-1

=\dfrac{1+2\sqrt{5}+5 }{8} -1

=\dfrac{6+2\sqrt{5}-8 }{8}

=\dfrac{\sqrt{5}-1 }{4}

Therefore,

3\cos\dfrac{2\pi}{5}-\cos\dfrac{\pi}{5}

=3\cdotp \dfrac{\sqrt{5}-1 }{4} -\dfrac{1+\sqrt{5} }{4}

=\dfrac{2\sqrt{5}-4 }{4}

=\dfrac{1}{2}\sqrt{5}-1


Steven Zheng Steven Zheng posted 6 days ago

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