Q81 / 180 · KCET 2013 · Mathematics

$\log \left(\sin 1^{\circ}\right) \cdot \log \left(\sin 2^{\circ}\right) \cdot \log \left(\sin 3^{\circ}\right)$ $\ldots \log \left(\sin 179^{\circ}\right)$

  • A
    is positive
  • B
    is negative
  • C
    lies between 1 and 180
  • Correct answer
    D
    is zero

Solution

$\log \left(\sin 1^{\circ}\right) \cdot \log \left(\sin 2^{\circ}\right) \cdot \log \left(\sin 3^{\circ}\right) \ldots \log \left(\sin 179^{\circ}\right)$ $=\log \sin 1^{\circ} \cdot \log \sin 2^{\circ} \ldots \log \sin 90^{\circ} \ldots \log \sin 179^{\circ}$ $=\log \sin 1^{\circ} \cdot \log \sin 2^{\circ} \ldots \log (1) \ldots \log \sin 179^{\circ}$ $\left.=\log \sin 1^{\circ} \cdot \log \sin 2 \ldots 0 \ldots \sin 90^{\circ}=1\right)$ $=0 \quad\left(\because \log \sin 179^{\circ} 1=0\right)$

Hence, the correct answer is option: 'D'.

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