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

Problem

tan(x)=−√(,3)/3

Solution

1. Identify the reference angle by finding the value of θ such that tan(θ)=√(,3)/3 in the first quadrant.

θ=π/6

2. Determine the quadrants where the tangent function is negative. Tangent is negative in Quadrant II and Quadrant IV.

Quadrant II and Quadrant IV

3. Calculate the specific solutions within one period [0,π) or the standard interval [0,2*π) In Quadrant II, the angle is π−π/6

x=(5*π)/6

4. Apply the periodicity of the tangent function. Since the period of tan(x) is π all solutions can be expressed by adding integer multiples of π

x=(5*π)/6+n*π

5. Define the variable n as any integer to complete the general solution.

n∈ℤ

Final Answer

tan(x)=−√(,3)/3⇒x=(5*π)/6+n*π

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