Tangent Is y over x

$\tan\left(\theta\right) = \frac{\sin\left(\theta\right)}{\cos\left(\theta\right)}$, so on the unit circle tangent is y divided by x: rise over run, the slope of the radius to the point.

At 30° the point is (√3/2, 1/2), so $\tan\left(30^{\circ}\right) = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}$, which is 1/√3, or $\frac{\sqrt{3}}{3}$. At 135° the point is (−√2/2, √2/2): equal sizes, opposite signs, so tan 135° = −1, a radius sloping down to the right.

For other angles use the same two questions as before: tan at the reference angle gives the size (√3/3, 1 or √3), the quadrant gives the sign. Tangent is positive in I and III, where across and height have the same sign.