Simplify tan((2pi)/3)

Problem

tan((2*π)/3)

Solution

1. Identify the reference angle for (2*π)/3 by subtracting it from π

Reference Angle=π−(2*π)/3=π/3

2. Determine the quadrant of the angle (2*π)/3

π/2<(2*π)/3<π

The angle is in Quadrant II.

3. Determine the sign of the tangent function in Quadrant II.
   In Quadrant II, sin(θ) is positive and cos(θ) is negative, so tan(θ) is negative.

4. Evaluate the tangent of the reference angle using known trigonometric values.

tan(π/3)=√(,3)

5. Combine the value and the sign to find the final result.

tan((2*π)/3)=−tan(π/3)=−√(,3)

Final Answer

tan((2*π)/3)=−√(,3)

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