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

Problem

sin(x/2)=−√(,3)/2

Solution

1. Identify the reference angle for the sine function. We know that sin(θ)=√(,3)/2 when 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 x/2 using the unit circle.

x/2=π+π/3+2*k*π

x/2=2*π−π/3+2*k*π

4. Simplify the expressions for the argument x/2 where k is any integer.

x/2=(4*π)/3+2*k*π

x/2=(5*π)/3+2*k*π

5. Solve for x by multiplying both sides of each equation by 2

x=(8*π)/3+4*k*π

x=(10*π)/3+4*k*π

Final Answer

sin(x/2)=−√(,3)/2⇒x=(8*π)/3+4*k*π,(10*π)/3+4*k*π

Want more problems? Check here!