Solve for x sin(x)cos(2x)+cos(x)sin(2x)=0

Problem

sin(x)*cos(2*x)+cos(x)*sin(2*x)=0

Solution

1. Identify the trigonometric identity for the sine of a sum, which states sin(A+B)=sin(A)*cos(B)+cos(A)*sin(B)

2. Apply the formula by letting A=x and B=2*x to condense the left side of the equation.

sin(x+2*x)=0

3. Simplify the expression inside the sine function.

sin(3*x)=0

4. Solve for the argument 3*x by identifying where the sine function equals zero.

3*x=n*π

where n is any integer (n∈ℤ.

5. Isolate x by dividing both sides by 3.

x=(n*π)/3

Final Answer

sin(x)*cos(2*x)+cos(x)*sin(2*x)=0⇒x=(n*π)/3,n∈ℤ

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