To solve the trigonometric equation 2sin(π-3x) + cos(2π-3x) = 0, we can use trigonometric identities to simplify the expression and find the values of x that satisfy the equation.
First, we will use the angle difference identity for sine: sin(π-3x) = sinπcos3x - cosπsin3x. Since sinπ = 0 and cosπ = -1, this simplifies to -cos3x.
Next, we will use the angle difference identity for cosine: cos(2π-3x) = cos2πcos3x + sin2πsin3x. Since cos2π = 1 and sin2π = 0, this simplifies to cos3x.
Substitute these simplified expressions back into the original equation:
Now, we solve for cos3x = 0. Since cos(90) = 0, this means that 3x = 90 + 360n or 3x = 270 + 360n, where n is an integer.
Solving for x, we get x = 30 + 120n or x = 90 + 120n, where n is an integer. Thus, the general solution for the equation 2sin(π-3x) + cos(2π-3x) = 0 is x = 30° + 120°n or x = 90° + 120°n, where n is an integer.
To solve the trigonometric equation 2sin(π-3x) + cos(2π-3x) = 0, we can use trigonometric identities to simplify the expression and find the values of x that satisfy the equation.
First, we will use the angle difference identity for sine: sin(π-3x) = sinπcos3x - cosπsin3x. Since sinπ = 0 and cosπ = -1, this simplifies to -cos3x.
Next, we will use the angle difference identity for cosine: cos(2π-3x) = cos2πcos3x + sin2πsin3x. Since cos2π = 1 and sin2π = 0, this simplifies to cos3x.
Substitute these simplified expressions back into the original equation:
2(-cos3x) + cos3x = 0
-2cos3x + cos3x = 0
-cos3x = 0
Now, we solve for cos3x = 0. Since cos(90) = 0, this means that 3x = 90 + 360n or 3x = 270 + 360n, where n is an integer.
Solving for x, we get x = 30 + 120n or x = 90 + 120n, where n is an integer. Thus, the general solution for the equation 2sin(π-3x) + cos(2π-3x) = 0 is x = 30° + 120°n or x = 90° + 120°n, where n is an integer.