Final answer:
The exact value of tan(210°) is negative because in the third quadrant, both sine and cosine are negative. The reference angle is 30°, and tan(30°) is \(√3/3\), so tan(210°) is -√3/3.
Step-by-step explanation:
To find the exact value of tan(210°), we look at the unit circle and recognize that 210° is in the third quadrant, where both sine and cosine are negative. The reference angle for 210° is 30°, as it is 180° + 30°. The tangent function is the ratio of sine to cosine, and for a reference angle of 30°, we have:
- sin(30°) = 1/2
- cos(30°) = \(√3/2\)
Therefore, the tangent of the reference angle is sin/cos = (1/2) / (\(√3/2\)) = 1/\(√3\) = \(√3/3\). But since we are in the third quadrant, where both sine and cosine are negative, the tangent value should also be negative. Thus, the exact value of tan(210°) is √3/3.