Question:

The value of sin5π12sinπ12\sin \frac{5\pi}{12} \sin \frac{\pi}{12} is

Updated On: Apr 20, 2024
  • 0
  • 12\frac{1}{2}
  • 1
  • 14\frac{1}{4}
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The Correct Option is D

Solution and Explanation

We can use the trigonometric identity sin(A+B)=sin(A)cos(B)+cos(A)sin(B)\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)
sin(5π12)sin(π12)=sin(π4+π6)sin(π12)\sin\left(\frac{5\pi}{12}\right) \sin\left(\frac{\pi}{12}\right) = \sin\left(\frac{\pi}{4} + \frac{\pi}{6}\right) \sin\left(\frac{\pi}{12}\right)
Using the identity, we have: 
sin(π4+π6)sin(π12)=(sin(π4)cos(π6)+cos(π4)sin(π6))sin(π12)\sin\left(\frac{\pi}{4} + \frac{\pi}{6}\right) \sin\left(\frac{\pi}{12}\right) = \left(\sin\left(\frac{\pi}{4}\right)\cos\left(\frac{\pi}{6}\right) + \cos\left(\frac{\pi}{4}\right)\sin\left(\frac{\pi}{6}\right)\right) \sin\left(\frac{\pi}{12}\right)

Simplifying further: 
(sin(π4)cos(π6)+cos(π4)sin(π6))sin(π12)=(1232+1212)sin(π12)(\sin(\frac{\pi}{4})\cos(\frac{\pi}{6}) + \cos(\frac{\pi}{4})\sin(\frac{\pi}{6}))\sin(\frac{\pi}{12}) = (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} \cdot \frac{1}{2})\sin(\frac{\pi}{12})
Combining terms:
(1232+1212)sin(π12)=(3+12)sin(π12)\left(\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} \cdot \frac{1}{2}\right)\sin\left(\frac{\pi}{12}\right) = \left(\frac{\sqrt{3} + 1}{2}\right) \cdot \sin\left(\frac{\pi}{12}\right)

Now, sin(π12)\sin\left(\frac{\pi}{12}\right) can be simplified using the half-angle formula: 
sin(π12)=1cos(π6)2=1322\sin\left(\frac{\pi}{12}\right) = \sqrt{\frac{1 - \cos\left(\frac{\pi}{6}\right)}{2}} = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} 

Substituting back into the expression: 
(3+12)sin(π12)=(3+12)1322\left(\frac{\sqrt{3} + 1}{2}\right) \sin\left(\frac{\pi}{12}\right) = \left(\frac{\sqrt{3} + 1}{2}\right) \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}}
Simplifying the expression:
 (3+12)1322=(3+12)234=(3+12)(232)\left(\frac{\sqrt{3} + 1}{2}\right) \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} = \left(\frac{\sqrt{3} + 1}{2}\right) \sqrt{\frac{2 - \sqrt{3}}{4}} = \left(\frac{\sqrt{3} + 1}{2}\right) \left(\frac{\sqrt{2} - \sqrt{3}}{2}\right)

Multiplying the fractions:
(3+12)(232)=63+214\left(\frac{\sqrt{3} + 1}{2}\right) \left(\frac{\sqrt{2} - \sqrt{3}}{2}\right) = \frac{\sqrt{6} - \sqrt{3} + \sqrt{2} - 1}{4}
Therefore, the value of sin(5π12)sin(π12)=63+214\sin\left(\frac{5\pi}{12}\right) \sin\left(\frac{\pi}{12}\right) = \frac{\sqrt{6} - \sqrt{3} + \sqrt{2} - 1}{4}
Hence, the correct option is (D) 14\frac{1}{4}.

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