To solve this trigonometric equation, we will manipulate the left side of the equation using trigonometric identities.
Starting with the given equation:
2sin^2(x) - cos^2(x) = sin(x)cos(x)
We know that sin^2(x) + cos^2(x) = 1 (Pythagorean identity).
Substitute sin^2(x) = 1 - cos^2(x) into the equation:
2(1 - cos^2(x)) - cos^2(x) = sin(x)cos(x)
Expanding the left side:
2 - 2cos^2(x) - cos^2(x) = sin(x)cos(x)2 - 3cos^2(x) = sin(x)cos(x)
Rearrange the equation:
3cos^2(x) = 2 - sin(x)cos(x)
Now, we need to use the identity sin(2x) = 2sin(x)cos(x) to solve the equation.
Substitute sin(2x) into the equation:
3cos^2(x) = 2 - sin(2x)
Since sin(2x) = 2sin(x)cos(x), we get:
3cos^2(x) = 2 - 2sin(x)cos(x)
Adding 2sin(x)cos(x) to both sides:
3cos^2(x) + 2sin(x)cos(x) = 2
Hence, the equation is now in the form desired.
To solve this trigonometric equation, we will manipulate the left side of the equation using trigonometric identities.
Starting with the given equation:
2sin^2(x) - cos^2(x) = sin(x)cos(x)
We know that sin^2(x) + cos^2(x) = 1 (Pythagorean identity).
Substitute sin^2(x) = 1 - cos^2(x) into the equation:
2(1 - cos^2(x)) - cos^2(x) = sin(x)cos(x)
Expanding the left side:
2 - 2cos^2(x) - cos^2(x) = sin(x)cos(x)
2 - 3cos^2(x) = sin(x)cos(x)
Rearrange the equation:
3cos^2(x) = 2 - sin(x)cos(x)
Now, we need to use the identity sin(2x) = 2sin(x)cos(x) to solve the equation.
Substitute sin(2x) into the equation:
3cos^2(x) = 2 - sin(2x)
Since sin(2x) = 2sin(x)cos(x), we get:
3cos^2(x) = 2 - 2sin(x)cos(x)
Adding 2sin(x)cos(x) to both sides:
3cos^2(x) + 2sin(x)cos(x) = 2
Hence, the equation is now in the form desired.