Find the Exact Value tan((14pi)/3)

Problem

tan((14*π)/3)

Solution

1. Find the coterminal angle by subtracting multiples of 2*π (which is (6*π)/3 until the angle is within the interval [0,2*π)

(14*π)/3−2*π=(14*π)/3−(6*π)/3=(8*π)/3

(8*π)/3−2*π=(8*π)/3−(6*π)/3=(2*π)/3

2. Identify the quadrant for the angle (2*π)/3 Since π/2<(2*π)/3<π the angle is in Quadrant II.

3. Determine the reference angle for (2*π)/3 in Quadrant II using the formula π−θ

π−(2*π)/3=π/3

4. Apply the tangent value for the reference angle π/3

tan(π/3)=√(,3)

5. Determine the sign of the tangent function in Quadrant II. Since tangent is negative in Quadrant II, the result must be negative.

tan((2*π)/3)=−√(,3)

Final Answer

tan((14*π)/3)=−√(,3)

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