Question

$2 \sin \binom{5\pi}{12} \times \cos \binom{7\pi}{12} =$
A) 1/√2
B) -1/2
C) 1/2
D) 1​

Answer 1

Given:-

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$\bf \implies2 \sin \binom{5\pi}{12} \times \cos \binom{7\pi}{12} = \:?$

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To Find:-

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$\bf \implies The \: Solution$

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STEP BY STEP EXPLANATION:-

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$\bf \implies2 \times \frac{1}{2} \times ( \sin(\pi) + \sin( \frac{ - \pi}{6} ))$

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$\bf \implies2 \times \frac{1}{2} \times ( 0 \sin( \frac{ - \pi}{6} ))$

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$\bf \implies2 \times \frac{1}{2} \times ( 0 - \frac{1}{2} )$

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$\bf \implies2 \times \frac{1}{2} \times ( - \frac{1}{2} )$

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$\bf \implies - 2 \times \frac{1}{2} \times \frac{1}{2}$

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$\bf \implies \cancel{- 2 }\times \frac{1}{2 \: this \: is \: cut} \times ( - \frac{1}{2} )$

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$\bf \implies - \frac{1}{2}$

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$\sf ∴ hence,\: b \: option \: is \: correct \: -\frac{1}{2}$

Answer 2

To find -

Find the value of -

2 sin [ 5 π / 12 ] × cos [ 7 π / 12 ]

Solution -

2 sin [ 5 π / 12 ] × cos [ 7 π / 12 ]

Here π radians = 180°

=> 2 sin [ 5 × 180° / 12 ] × cos [ 7 × 180° / 12 ]

=> 2 sin [ 900 ° / 12 ] × cos [ 1260° / 12 ]

=> 2 sin [ 75 ° ] × cos [ 105 ° ]

=> 2 sin [ 30° + 45 ° ] × cos [ 105 ° ]

Sin ( 30° + 45° )

=> sin30° × cos45° + sin45° × cos30°

=> [ √2 + √6 ] / 4

=> √2 ( 1 + √3 ) / 4

Substituting -

=> 2 × [ √2 + √6 / 4 ] × cos 105°

=> ½ [ √2 + √6 ] × cos 105°

Cos 105°

=> cos ( 60° + 45° )

=> cos 60° cos 45° - sin 60° sin 45°

=> √2 ( 1 - √3 ) / 4

Simplifying

=> √2 ( 1 + √3 ) / 4 × √2 ( 1 - √3 ) / 4

=> 2 [ 1 - 3 ] / 16

=> [ -2 / 4 ]

=> -½ .

Option B is the correct answer .

This is the required answer.

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