Solve for x 2sin(x)^2+ square root of 3sin(x)=0

Problem

2*sin2(x)+√(,3)*sin(x)=0

Solution

1. Identify the common factor in the terms of the equation. Both terms contain sin(x)

2. Factor out the common term sin(x) from the expression.

sin(x)*(2*sin(x)+√(,3))=0

3. Apply the zero product property by setting each factor equal to zero.

sin(x)=0

2*sin(x)+√(,3)=0

4. Solve the first equation for x The sine function is zero at integer multiples of π

x=n*π

5. Isolate the sine term in the second equation.

2*sin(x)=−√(,3)

sin(x)=−√(,3)/2

6. Solve for x in the second equation using the unit circle. The sine function equals −√(,3)/2 at (4*π)/3 and (5*π)/3 within the interval [0,2*π)

x=(4*π)/3+2*n*π

x=(5*π)/3+2*n*π

Final Answer

x=n*π,(4*π)/3+2*n*π,(5*π)/3+2*n*π

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