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Question

Mathematics Question on Trigonometric Functions

log(sin⁡ 1∘)×log⁡(sin⁡ 2∘)×log⁡(sin⁡ 3∘)…log(sin 179∘)log (\sin \,1^\circ ) \times \log(\sin\, 2^\circ) \times \log(\sin \,3^\circ)\dots log (sin \,179^{\circ})

A

is positive

B

is negative

C

lies between 1 and 180

D

is zero

Answer

is zero

Explanation

Solution

log⁡(sin⁡1∘)⋅log⁡(sin⁡2∘)⋅log⁡(sin⁡3∘)…log⁡(sin⁡179∘)\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∘⋅log⁡sin⁡2∘…log⁡sin⁡90∘…log⁡sin⁡179∘=\log \sin 1^{\circ} \cdot \log \sin 2^{\circ} \ldots \log \sin 90^{\circ} \ldots \log \sin 179^{\circ}
=log⁡sin⁡1∘⋅log⁡sin⁡2∘…log⁡(1)…log⁡sin⁡179∘     (∵sin⁡90∘=1)= \log \sin 1^{\circ} \cdot \log \sin 2^{\circ} \ldots \log (1) \ldots \log \sin 179^{\circ} \,\,\,\,\,\left(\because \sin 90^{\circ}=1\right)
=log⁡sin⁡1∘⋅log⁡sin⁡2…0…log⁡sin⁡179∘= \log \sin 1^{\circ} \cdot \log \sin 2 \ldots 0 \ldots \log \sin 179^{\circ}
(∵ log 1=0)(\because \,log \,1=0)
=0= 0