Find the Exact Value tan((5pi)/6)

Problem

tan((5*π)/6)

Solution

1. Identify the reference angle by finding the distance from the angle (5*π)/6 to the nearest x-axis, which is π

Reference Angle=π−(5*π)/6=π/6

2. Determine the quadrant of the angle (5*π)/6 Since π/2<(5*π)/6<π the angle lies in Quadrant II.

3. Determine the sign of the tangent function in Quadrant II. In this quadrant, sine is positive and cosine is negative, so tangent is negative.

4. Evaluate the tangent of the reference angle π/6 using known trigonometric values.

tan(π/6)=1/√(,3)

5. Apply the negative sign to the reference value to find the exact value for the original angle.

tan((5*π)/6)=−1/√(,3)

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

−1/√(,3)⋅√(,3)/√(,3)=−√(,3)/3

Final Answer

tan((5*π)/6)=−√(,3)/3

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