Find the Exact Value tan(-(7pi)/3)

Problem

tan(−(7*π)/3)

Solution

1. Find a coterminal angle by adding multiples of 2*π to the angle until it falls within the standard interval [0,2*π)

−(7*π)/3+2*π=−(7*π)/3+(6*π)/3=−π/3

−π/3+2*π=−π/3+(6*π)/3=(5*π)/3

2. Identify the reference angle for (5*π)/3 which is located in the fourth quadrant.

(θ_ref)=2*π−(5*π)/3=π/3

3. Determine the sign of the tangent function in the fourth quadrant. Since tan(θ)=sin(θ)/cos(θ) and sine is negative while cosine is positive in the fourth quadrant, the result will be negative.

tan((5*π)/3)=−tan(π/3)

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

tan(π/3)=√(,3)

5. Combine the sign and value to find the final result.

tan(−(7*π)/3)=−√(,3)

Final Answer

tan(−(7*π)/3)=−√(,3)

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