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

Problem

sin(2*x)*cos(x)+cos(2*x)*sin(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 to the left side of the equation by letting A=2*x and B=x

3. Simplify the expression inside the sine function.

sin(2*x+x)=0

sin(3*x)=0

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

3*x=n*π

5. Isolate x by dividing both sides by 3, where n is any integer.

x=(n*π)/3

Final Answer

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

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