Tangent Is y over x

tan(θ) = sin(θ)/cos(θ), 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(30°) = (1/2)/(√(3)/2), which is 1/√3, or √(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.