Tangent is sine over cosine: opposite over adjacent in a right triangle, and y over x on the unit circle. The 30-60-90 and 45-45-90 triangles give exact sine, cosine and tangent for π/6, π/4 and π/3. The unit circle then gives every reflection: π - x keeps sine, π + x keeps tangent, 2π - x keeps cosine. Cosine is even and sine and tangent are odd, and all three repeat, sine and cosine every 2π and tangent every π. The crew reads the mast's shadow with all of it.