A tower is 50 m high, and from a point on level ground the angle of elevation of its top is 75°. Taking tan75° = 2 + √3, the distance of that point from the foot of the tower is:

tan75° = 50/d, so d = 50/(2 + √3). Multiplying above and below by 2 − √3 and using (2 + √3)(2 − √3) = 4 − 3 = 1 leaves d = 50(2 − √3) ≈ 13.4 m — suitably short, since a 75° sighting is steep. Multiplying by the tangent instead of dividing gives 50(2 + √3) ≈ 186.6 m, rationalising with the wrong sign gives the negative 50(√3 − 2), and 50/√3 ≈ 28.9 m answers the question for 60°.