Find the Value Using the Unit Circle tan((5pi)/6)

Problem

tan((5*π)/6)

Solution

1. Identify the angle (5*π)/6 on the unit circle. This angle is located in the second quadrant because π/2<(5*π)/6<π

2. Determine the coordinates (x,y) for the angle (5*π)/6 On the unit circle, these coordinates are:

x=cos((5*π)/6)=−√(,3)/2

y=sin((5*π)/6)=1/2

3. Apply the formula for the tangent function, which is the ratio of the ycoordinate to the xcoordinate:

tan(θ)=y/x

4. Substitute the values into the formula:

tan((5*π)/6)=1/2/(−√(,3)/2)

5. Simplify the fraction by multiplying by the reciprocal of the denominator:

tan((5*π)/6)=1/2⋅(−2/√(,3))

tan((5*π)/6)=−1/√(,3)

6. Rationalize the denominator by multiplying the numerator and denominator by √(,3)

tan((5*π)/6)=−√(,3)/3

Final Answer

tan((5*π)/6)=−√(,3)/3

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