Solve for x sin(2x)=- square root of 3sin(x)

Problem

sin(2*x)=−√(,3)*sin(x)

Solution

1. Apply the double angle identity for sine, which states sin(2*x)=2*sin(x)*cos(x) to rewrite the left side of the equation.

2*sin(x)*cos(x)=−√(,3)*sin(x)

2. Move all terms to one side by adding √(,3)*sin(x) to both sides to set the equation equal to zero.

2*sin(x)*cos(x)+√(,3)*sin(x)=0

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

sin(x)*(2*cos(x)+√(,3))=0

4. Set each factor to zero using the zero product property to find the possible solutions for x

sin(x)=0

2*cos(x)+√(,3)=0

5. Solve the first equation sin(x)=0 The sine function is zero at integer multiples of π

x=n*π

6. Solve the second equation by isolating the cosine term.

cos(x)=−√(,3)/2

7. Determine the angles where the cosine is −√(,3)/2 This occurs in the second and third quadrants.

x=(5*π)/6+2*n*π

x=(7*π)/6+2*n*π

Final Answer

sin(2*x)=−√(,3)*sin(x)⇒x=n*π,(5*π)/6+2*n*π,(7*π)/6+2*n*π

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