Question

[30 pontos] Resolva em R a equação

Answer 1

sen [ x + ( π/4 ) ] + cos [ x + ( π/4 ) ] = ( √2 )/2 → divida tudo por √2 

( 1/√2 ).sen [ x + ( π/4 ) ] + ( 1/√2 ).cos [ x + ( π/4 ) ] = ( √2 )/2√2 

[ (√2 )/2 ].sen [ x + ( π/4 ) ] + [ (√2 )/2 ].cos [ x + ( π/4 ) ] = 1/2 

sen ( π/4 ).sen [ x + ( π/4 ) ] + cos ( π/4 ).cos [ x + ( π/4 ) ] = 1/2 

Que equivale à: 

cos [ x + ( π/4 ) ].cos ( π/4 ) + sen [ x + ( π/4 ) ].sen ( π/4 ) 


Recordando que cos (a).cos (b) + sen (a).sen (b) = cos ( a - b ), temos; 


cos [ x + ( π/4 ) - ( π/4 ) ] = 1/2 

cos ( x ) = 1/2 


Como 1/2 = cos [ ( π/3 ) + 2kπ ] = sen [ ( 5π/3 ) + 2kπ ] , com k Є Z , podemos escrever: 

x = ± ( π/3 ) + 2kπ, k Є Z. 

R ──────► S = { x Є IR/ x = ± ( π/3 ) + 2kπ, k Є Z }