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avatar+9675 

cos4x(sin2x)(sinx)(cosx)2cos2x+2sin2x+1=0

 Aug 31, 2016

Best Answer 

 #3
avatar
+5

Solve for x:
1-2 cos^2(x)+cos^4(x)+2 sin^2(x)-cos(x) sin(x) sin(2 x) = 0

 

Reduce trigonometric functions:
-3/8 (-3+4 cos(2 x)-cos(4 x)) = 0

 

Multiply both sides by -8/3:
-3+4 cos(2 x)-cos(4 x) = 0

 

Transform -3+4 cos(2 x)-cos(4 x) into a polynomial with respect to cos(2 x) using cos(4 x) = 2 cos^2(2 x)-1:
-2+4 cos(2 x)-2 cos^2(2 x) = 0

 

Divide both sides by -2:
1-2 cos(2 x)+cos^2(2 x) = 0

 

Write the left hand side as a square:
(cos(2 x)-1)^2 = 0

 

Take the square root of both sides:
cos(2 x)-1 = 0

 

Add 1 to both sides:
cos(2 x) = 1

 

Take the inverse cosine of both sides:
2 x = 2 π n for n element Z

 

Divide both sides by 2:
Answer: |x = π n for n element Z

 Aug 31, 2016
 #1
avatar+1084 
0

Thats tough

 Aug 31, 2016
 #2
avatar+9675 
0

Of course, I made this question up! :P There is still a solution, that was not a prank.

MaxWong  Aug 31, 2016
 #3
avatar
+5
Best Answer

Solve for x:
1-2 cos^2(x)+cos^4(x)+2 sin^2(x)-cos(x) sin(x) sin(2 x) = 0

 

Reduce trigonometric functions:
-3/8 (-3+4 cos(2 x)-cos(4 x)) = 0

 

Multiply both sides by -8/3:
-3+4 cos(2 x)-cos(4 x) = 0

 

Transform -3+4 cos(2 x)-cos(4 x) into a polynomial with respect to cos(2 x) using cos(4 x) = 2 cos^2(2 x)-1:
-2+4 cos(2 x)-2 cos^2(2 x) = 0

 

Divide both sides by -2:
1-2 cos(2 x)+cos^2(2 x) = 0

 

Write the left hand side as a square:
(cos(2 x)-1)^2 = 0

 

Take the square root of both sides:
cos(2 x)-1 = 0

 

Add 1 to both sides:
cos(2 x) = 1

 

Take the inverse cosine of both sides:
2 x = 2 π n for n element Z

 

Divide both sides by 2:
Answer: |x = π n for n element Z

Guest Aug 31, 2016
 #5
avatar+9675 
0

No wonder you gave me a 1 star guys, I made a mistake

The correct equation is: cos4x(sin2x)(sinx)(cosx)2cos2x+sin4x+2sin2x+1=0

 Sep 1, 2016
 #6
avatar+9675 
0

I am really sorry :(

MaxWong  Sep 1, 2016
 #7
avatar+9675 
0

Also, tips!! 

Tips: Please do NOT solve it as a normal trig equation!! It can actually be solved like a quadratic!!

MaxWong  Sep 1, 2016
 #8
avatar+118696 
+5

Hi Max :)

 

\cos^4x-(\sin 2x)(\sin x)(\cos x)-2\cos^2x+\sin^4x+2\sin^2x+1=0

 

cos4x(sin2x)(sinx)(cosx)2cos2x+sin4x+2sin2x+1=0cos4x+sin4x(sin2x)(sin2x2)2(cos2xsin2x)+1=0cos4x+sin4x+2cox2xsin2x2cox2xsin2x(sin2x)(sin2x2)2(cos2x)+1=0(cos2x+sin2x)24cox2xsin2x2(sin2x)(sin2x2)2(cos2x)+1=01(2cosxsinx)22(sin2x)(sin2x2)2(cos2x)+1=0(sin2x)22(sin2x)222(cos2x)+2=0(sin2x)22(cos2x)+2=0[1cos2(2x)]2(cos2x)+2=01+(cos2x)22(cos2x)+2=0(cos2x)22(cos2x)+1=0lety=cos2xy22y+1=0(y1)2=0y=1cos2x=12x=2πnnZx=πnnZ

 

check

cos4(2πn)(sin2(2πn))(sin(2πn))(cos(2πn))2cos2(2πn)+sin4(2πn)+2sin2(2πn)+1=0102+0+0+1=0GOOD

 Sep 1, 2016

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