Question

Determine the value of \sin 10\degree\cdotp \sin 30\degree\cdotp\sin 50\degree\cdotp\sin 70\degree

Collected in the board: Trigonometry

Steven Zheng posted 2 months ago


Answer

\sin 10\degree\cdotp \sin 30\degree\cdotp\sin 50\degree\cdotp\sin 70\degree

=\dfrac{2\sin 10\degree \cos 10\degree\cdotp \sin 30\degree\cdotp\sin 50\degree\cdotp\sin(90\degree -20 \degree ) }{2 \cos 10\degree}

=\dfrac{2\sin 20\degree\cos 20\degree\cdotp \sin 30\degree\cdotp\sin (90\degree-40\degree ) }{4 \cos 10\degree}

=\dfrac{2\sin 40\degree\cos 40\degree \cdotp \sin 30\degree }{8\cos 10\degree}

=\dfrac{\sin 80 \degree \sin 30\degree }{8\cos 10\degree}

=\dfrac{1}{16}

Steven Zheng posted 2 months ago

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