Sin 3x + COS 3x=0

To solve the equation sin(3x) + cos(3x) = 0, we can use trigonometric identities to rewrite it in terms of a single trigonometric function.

1. Start by using the angle addition formula for sine and cosine: sin(3x) + cos(3x) = sin(3x) + sin(90° - 3x)

2. By the sum-to-product formula for sine: sin(a) + sin(b) = 2 sin((a+b)/2) cos((a-b)/2) sin(3x) + sin(90° - 3x) = 2 sin((3x + 90° - 3x)/2) cos((3x - 90° + 3x)/2) sin(3x) + sin(90° - 3x) = 2 sin(45°) cos(3x - 45°)

3. Since sin(45°) = √2/2, we have: sin(3x) + cos(3x) = 2(√2/2)cos(3x - 45°) = √2cos(3x - 45°)

Now, we have sin(3x) + cos(3x) = √2cos(3x - 45°) = 0

To solve this equation: √2cos(3x - 45°) = 0 cos(3x - 45°) = 0 3x - 45° = 90° + 180°n or 3x - 45° = 270° + 180°n 3x = 135° + 180°n or 3x = 315° + 180°n x = 45° + 60°n or x = 105° + 60°n

Therefore, the solutions to the equation sin(3x) + cos(3x) = 0 are x = 45° + 60°n and x = 105° + 60°n, where n is an integer.