Evaluate tan(210 degrees )

Problem

tan(210^∘)

Solution

1. Identify the reference angle by finding the difference between 210^∘ and the nearest x-axis, which is 180^∘

(θ_ref)=210^∘−180^∘=30^∘

2. Determine the quadrant of the angle 210^∘ which lies in the third quadrant (Quadrant III).

3. Apply the sign rule for the tangent function in the third quadrant, where tangent is positive.

tan(210^∘)=tan(30^∘)

4. Evaluate the exact value of tan(30^∘) using the unit circle or special triangles.

tan(30^∘)=1/√(,3)

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

1/√(,3)⋅√(,3)/√(,3)=√(,3)/3

Final Answer

tan(210^∘)=√(,3)/3

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