Solve by Factoring square root of 3cot(x)+1=0

Problem

√(,3)*cot(x)+1=0

Solution

1. Isolate the trigonometric term by subtracting 1 from both sides of the equation.

√(,3)*cot(x)=−1

2. Divide both sides by √(,3) to solve for cot(x)

cot(x)=−1/√(,3)

3. Rewrite the equation in terms of tan(x) using the reciprocal identity tan(x)=1/cot(x)

tan(x)=−√(,3)

4. Identify the reference angle. Since tan(π/3)=√(,3) the reference angle is π/3

5. Determine the quadrants where the tangent function is negative, which are Quadrant II and Quadrant IV.

x=π−π/3=(2*π)/3

x=2*π−π/3=(5*π)/3

6. Generalize the solution by adding multiples of the period of the tangent function, which is π

x=(2*π)/3+n*π

Final Answer

√(,3)*cot(x)+1=0⇒x=(2*π)/3+n*π

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