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

Problem

sin(2*x)=−√(,3)/2

Solution

1. Identify the reference angle for the sine function. We know that sin(θ)=√(,3)/2 when the reference angle is 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 2*x by applying the quadrant rules.

(θ_1)=π+π/3+2*π*n=(4*π)/3+2*π*n

(θ_2)=2*π−π/3+2*π*n=(5*π)/3+2*π*n

4. Set the argument 2*x equal to these general solutions.

2*x=(4*π)/3+2*π*n

2*x=(5*π)/3+2*π*n

5. Isolate x by dividing every term in the equations by 2

x=(2*π)/3+π*n

x=(5*π)/6+π*n

Final Answer

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

Want more problems? Check here!