Use an Addition or Subtraction Formula to simplify the equation.

sin (θ) cos (3θ) + cos (θ) sin (3θ) = 0

Find all solutions in the interval [0, 2π). (Enter your answers as a comma-separated list.)

Sin (A+B) = sin A cos B+cos A cos B

In this equation:

A=Ф

B=3 Ф

Therefore:

sin (Ф) cos (3 Ф) + cos (Ф) sin (3 Ф) = sin (Ф+3 Ф) = sin (4 Ф).

Then, we have:

sin (4 Ф) = 0

4 Ф=sin⁻¹ 0=kπ (k∈Z) Z = ... - 2,-1,0,1,2, ...

4 Ф=kπ

Ф=kπ/4

answer: Ф=Kπ/4 (k∈Z) Z = ... - 2,-1,0,1,2 ...

We can check it out our answer is right.

For example:

if Z=1; then Ф=π/4

sin (π/4) + cos (3π/4) + cos (π/4) sin (3π/4) = (√2 / 2) (-√2/2) + (√2/2) (√2/2) =

=-1/2+1/2=0