Question

evalute sin pi/12....................
​

Answer 1

Answer:

(1/4)($\sqrt{6}-\sqrt{2} )$

Step-by-step explanation:

We want to find replacement angles for  π /12  that will produce exact values

These must come from :  π/6 , π /3 , π /4

⇒ sin ( π /12 ) =sin ( π /3 − π /4 )

Using the appropriate addition formula  

sin ( A ± B ) = sin A cos B ± cos A sin B

⇒ sin ( π /3− π /4 )

= sin (π/3 ) cos( π/4 ) − cos (π /3 ) sin ( π /4 )

Extract  exact values from triangles  

sin ( π/ 3 ) = √ 3 /2 , sin ( π /4 ) = 1/ √ 2

and

cos ( π /3 ) = 1/ 2 , cos ( π /4 ) = 1 √ 2

now substitute into the right side of the expansion.

= √ 3 /2 × 1 /√ 2 − 1 /2 × 1 /√ 2

= √ 3 /2 √ 2 − 1 /2 √ 2

= (√ 3 − 1 )/(2 √ 2)

and rationalising the denominator  

gives  

( √ 3 − 1 ) × √ 2 /(2 √ 2 × √ 2)

= (√ 6 − √ 2 )/4

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