Question

Hei. Am nevoie de ajutor la acest exercițiu. Trebuie sa arat ca : sin^2(pi/8)+sin^2(3pi/8)+sin^2 (5pi/8)+sin^2(7pi/4) este egal cu 2. Mulțumesc

Answer 1

sin²π/8 +sin²3π/8+sin²5π/8+sin²7π4=
cos 2x= 1- 2sin²x
1 -2 sin² π/8= cos ( 2· π/8 ) = cos π/4 = √2 /2 
1- 2sin²3π/8=  cos ( 2· 3π/8) = cos 3π/4
1 -2sin²5π/8 = cos ( 2· 5π/8) = cos 5π/4
1- 2sin²7π/8= cos ( 2·7π/8)  = cos 7π/4
-------------------------------------------------------------- adunare
4 - 2 exercitiu=2cos [( 3π/4 +5π/4)/2] ·[ cos( 5π/4 - 3π/4)/2]  + 
+ 2cos[ ( 7π/4 + π/4)/2]·cos [( 7π/4 - π/4)/2]=
=2·cosπ·cosπ/4 + 2·cosπ·cos3π/4 =
       ↓                       ↓
       -1                       -1
 4 - 2 exercitiu = -2 ·√2/2 -2 ( -√2/2) 
4 - 2 exercitiu = 0
2 exercitiu =4 
ex =2