Find the Exact Value tan((8pi)/3)

Problem

tan((8*π)/3)

Solution

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

(8*π)/3−2*π=(8*π)/3−(6*π)/3

(8*π)/3−(6*π)/3=(2*π)/3

2. Determine the quadrant of the angle (2*π)/3 Since π/2<(2*π)/3<π the angle lies in Quadrant II.

3. Find the reference angle (θ_R) for an angle in Quadrant II using the formula (θ_R)=π−θ

(θ_R)=π−(2*π)/3

(θ_R)=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((8*π)/3)=−√(,3)

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