Sin 3x + COS 3x=0

To solve the equation sin(3x) + cos(3x) = 0, we can use the trigonometric identity for the sum of angles for sine and cosine.

sin(A + B) = sinA * cosB + cosA * sinB

In this case, A = B = 3x. Therefore:

sin(3x + 3x) = sin(6x) = sin(3x) * cos(3x) + cos(3x) * sin(3x)

Now substituting back into the original equation:

sin(6x) = 2 * sin(3x) * cos(3x)

Since sin(6x) = 0, then sin(3x) * cos(3x) = 0

This equation can be satisfied when either sin(3x) = 0 or cos(3x) = 0.

For sin(3x) = 0: 3x = k * π where k is an integer x = k * π / 3 since the period of sine function is 2π

For cos(3x) = 0: 3x = (2k + 1) * π / 2 where k is an integer x = (2k + 1) * π / 6 since the period of cosine function is 2π

Therefore, the general solutions for the equation sin(3x) + cos(3x) = 0 are: x = k * π / 3 for sin(3x) = 0 or x = (2k + 1) * π / 6 for cos(3x) = 0 where k is an integer.