Solve this equations. sqrt(3)cos 2x+3 sin 2x=3 x in [0, 2pi] cos^2 x-sin^2 x=sqrt(3) sin x cos x x in [0, 2pi]

Silvina2b

Silvina2b

Answered question

2022-08-05

Solve this equations.
3 cos 2 x + 3 sin 2 x = 3 , x [ 0 , 2 π ]
cos 2 x sin 2 x = 3 sin x cos x , x [ 0 , 2 π ]

Answer & Explanation

afinat4s

afinat4s

Beginner2022-08-06Added 13 answers

(i) 3 cos ( 2 x ) + 3 sin ( 2 x ) = 3...(1)
Let R cos ( 2 x α ) = 3, where R and α are positive constants.
R cos ( 2 x ) cos ( α ) + R sin ( 2 x ) sin ( α ) = 3...(2)
By comparing Eqn. (1) and Eqn. (2), we have
R cos ( α ) = 3 ....(3) and
R sin ( α ) = 3...(4)
( 3 ) 2 + ( 4 ) 2 , w e   h a v e   R 2 ( sin 2 α + cos 2 α ) = 3 + 9 = 12
R 2 = 12   a s   sin 2 α + cos 2 α = 1
R = 12
( 4 ) ( 3 ) , we have sin ( α ) cos ( α ) = tan ( α ) = 3 3
α = tan 1 ( 3 3 ) = π 3
The problem, (1), can be rewritten as 12 cos ( 2 x π 3 ) = 3
cos ( 2 x π 3 ) = 3 12 = 3 2 3 = 3 3 2 3 = 3 2
2 x π 3 = π 6 , 11 π 6 , 13 π 6 (choose angles so that x will bewithin ( 0 , 2 π ))
x = π 4 , 13 π 12 , 15 π 12
(ii) cos 2 ( x ) sin 2 ( x ) = 3 sin ( x ) cos ( x )
cos ( 2 x ) = 3 ( 1 2 sin ( 2 x ) ) ( sin ( 2 x ) = 2 sin ( x ) cos ( x ) )
sin ( 2 x ) cos ( 2 x ) = tan ( 2 x ) = 2 3
2 x = 49.1 , 229.1 , 409.1 , 589.1
x = 24.55 , 114.55 , 204.55 , 294.55

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