To solve this trigonometric equation, we can use the double angle formula for cosine:
2cos(x)cos(11π/2) - 2sin(x)sin(11π/2) = sin(x)
Now, we know that cos(11π/2) = cos(6π/2 + 5π/2) = cos(3π) = -1 and sin(11π/2) = sin(6π/2 + 5π/2) = sin(3π) = 0.
So, the equation simplifies to:
-2cos(x) - 0 = sin(x)
Which can be simplified further to:
-2cos(x) = sin(x)
Now, we can square both sides to eliminate the trigonometric functions:
(-2cos(x))^2 = (sin(x))^2
4cos^2(x) = sin^2(x)
Using the Pythagorean identity cos^2(x) + sin^2(x) = 1, we can substitute cos^2(x) = 1 - sin^2(x):
4(1 - sin^2(x)) = sin^2(x)
4 - 4sin^2(x) = sin^2(x)
4 = 5sin^2(x)
sin^2(x) = 4/5
Taking the square root of both sides, we get:
sin(x) = ± √(4/5)
Therefore, the solutions to the trigonometric equation are:
sin(x) = √(4/5) or sin(x) = -√(4/5)
To solve this trigonometric equation, we can use the double angle formula for cosine:
2cos(x)cos(11π/2) - 2sin(x)sin(11π/2) = sin(x)
Now, we know that cos(11π/2) = cos(6π/2 + 5π/2) = cos(3π) = -1 and sin(11π/2) = sin(6π/2 + 5π/2) = sin(3π) = 0.
So, the equation simplifies to:
-2cos(x) - 0 = sin(x)
Which can be simplified further to:
-2cos(x) = sin(x)
Now, we can square both sides to eliminate the trigonometric functions:
(-2cos(x))^2 = (sin(x))^2
4cos^2(x) = sin^2(x)
Using the Pythagorean identity cos^2(x) + sin^2(x) = 1, we can substitute cos^2(x) = 1 - sin^2(x):
4(1 - sin^2(x)) = sin^2(x)
4 - 4sin^2(x) = sin^2(x)
4 = 5sin^2(x)
sin^2(x) = 4/5
Taking the square root of both sides, we get:
sin(x) = ± √(4/5)
Therefore, the solutions to the trigonometric equation are:
sin(x) = √(4/5) or sin(x) = -√(4/5)