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

Problem

sin(3*x)=−√(,3)/2

Solution

1. Identify the reference angle for the sine function. The value sin(θ)=√(,3)/2 occurs at θ=π/3

2. Determine the quadrants where the sine function is negative. Sine is negative in the third and fourth quadrants.

3. Find the general solutions for the argument 3*x using the reference angle.

3*x=π+π/3+2*k*π=(4*π)/3+2*k*π

3*x=2*π−π/3+2*k*π=(5*π)/3+2*k*π

4. Solve for x by dividing the entire expression by 3 where k is any integer.

x=(4*π)/9+(2*k*π)/3

x=(5*π)/9+(2*k*π)/3

Final Answer

sin(3*x)=−√(,3)/2⇒x=(4*π)/9+(2*k*π)/3,(5*π)/9+(2*k*π)/3

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