In ancient times, it could be difficult to determine distances, especially for unreachable areas. For example, observers standing on a dock would have difficulty determining the distance to a ship. However with the application of trigonometry they were able to determine how far away the ship was.

The two observers stood at a set distance $d\=d_1\+d_2$ from each other. They then used their instruments to measure the angles ${\mathrm{\theta }}_1 and {\mathrm{\theta }}_2$to the boat. Now, the tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side, in this case:

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Therefore we know that $d \= \frac{L}{\tan {\mathrm{\theta }}_1}\+\frac{L}{\tan {\mathrm{\theta }}_2}$, or solving for $L$ and simplifying,  $L\=\frac{d\cdot \sin {\mathrm{\theta }}_1\cdot \sin {\mathrm{\theta }}_2}{\sin \left({\mathrm{\theta }}_1\+{\mathrm{\theta }}_2\right)}\.$