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Question
The value of `sin pi/10 sin (13pi)/10` is ______.
`["Hint: Use" sin18^circ = (sqrt5 - 1)/4 "and" cos36^circ = (sqrt5 + 1)/4]`
Options
`1/2`
`-1/2`
`-1/4`
1
Solution
The value of `sin pi/10 sin (13pi)/10` is `bbunderline(-1/4)`.
Explanation:
`sin pi/10 sin (13pi)/10 = sin pi/10 . sin(pi + (3pi)/10)`
= `sin pi/10 . (-sin (3pi)/10)`
= –sin 18° . sin 54°
= –sin 18° . sin(90° – 36°)
= –sin 18° . cos 36°
= `-((sqrt(5) - 1)/4)((sqrt(5) + 1)/4)` ......`[because sin18^circ = (sqrt(5) - 1)/4, cos 36^circ = (sqrt(5) + 1)/4]`
= `-((5 - 1)/16)`
= `-1/4`
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