Simplify tan((5pi)/6)

Problem

tan((5*π)/6)

Solution

1. Identify the reference angle for (5*π)/6 by finding its distance from the horizontal axis π

Reference Angle=π−(5*π)/6=π/6

2. Determine the quadrant in which the angle (5*π)/6 lies. Since π/2<(5*π)/6<π the angle is 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.

tan((5*π)/6)=−tan(π/6)

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

tan(π/6)=1/√(,3)

5. Rationalize the denominator of the result.

1/√(,3)=√(,3)/3

6. Combine the value with the correct sign determined in step 3.

tan((5*π)/6)=−√(,3)/3

Final Answer

tan((5*π)/6)=−√(,3)/3

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