Find the Value Using the Unit Circle tan((2pi)/3)

Problem

tan((2*π)/3)

Solution

1. Identify the angle θ=(2*π)/3 on the unit circle, which is located in the second quadrant.

2. Determine the coordinates (x,y) for the angle (2*π)/3 where x=cos((2*π)/3) and y=sin((2*π)/3)

3. Recall the unit circle values for this angle: x=−1/2 and y=√(,3)/2

4. Apply the formula for the tangent function, which is defined as the ratio of the ycoordinate to the xcoordinate.

tan(θ)=y/x

5. Substitute the known values into the ratio.

tan((2*π)/3)=√(,3)/2/(−1/2)

6. Simplify the fraction by multiplying by the reciprocal of the denominator.

tan((2*π)/3)=√(,3)/2⋅(−2/1)

tan((2*π)/3)=−√(,3)

Final Answer

tan((2*π)/3)=−√(,3)

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