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

Problem

tan(x/2)=−√(,3)

Solution

1. Identify the basic angle where the tangent function equals √(,3) Since tan(π/3)=√(,3) the reference angle is π/3

2. Determine the general solution for the tangent equation. The tangent function has a period of π so tan(θ)=y implies θ=arctan(y)+n*π for any integer n

3. Apply the inverse tangent to the given value. Since the value is −√(,3) we use arctan(−√(,3))=−π/3

4. Set up the equation for the argument of the tangent function.

x/2=−π/3+n*π

5. Isolate x by multiplying both sides of the equation by 2

x=2*(−π/3+n*π)

6. Distribute the 2 to find the final general solution.

x=−(2*π)/3+2*n*π

Final Answer

tan(x/2)=−√(,3)⇒x=−(2*π)/3+2*n*π

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